An Account of the Calculations Made from the Survey and Measures Taken at Schehallien, in Order to Ascertain the Mean Density of the Earth. By Charles Hutton, Esq. F. R. S.
Author(s)
Charles Hutton
Year
1778
Volume
68
Pages
108 pages
Language
en
Journal
Philosophical Transactions of the Royal Society of London
Full Text (OCR)
XXXIII. An Account of the Calculations made from the Survey and Measures taken at Schehallien, in order to ascertain the mean Density of the Earth. By Charles Hutton, Esq. F. R. S.
Read May 21, 1778.
THE survey from which these calculations have been made was taken at and about the hill Schehallien in Perthshire, in the years 1774, 1775, and 1776, by the direction, and partly under the inspection, of the Rev. Nevil Maskelyne, D.D. F.R.S. and Astronomer Royal, by whom the manner of making the survey has already been fully explained in the Philosophical Transactions for 1775.
I have therefore only to give an account of the measures of the lines and angles, and of the calculations which I have raised from them with all possible care and faithfulness, for the purpose of determining the measure of the ratio of the mean density of the earth to that of water or any other known matter.
These calculations were naturally and unavoidably long and tedious; and the more so as the business was in a manner quite new, which laid me under the necessity
Mr. Hutton's Calculations to ascertain
of inventing and describing such modes of computation as should be proper to be applied in so important and delicate a business. Having, at length, with close and unwearied application for a considerable time completed all the calculations; I have, in the following sheets, drawn up an account of those operations, with the results arising from them; and have accompanied them with such drawings as are necessary to illustrate the descriptions. I have also inserted a synopsis of the measures which were taken of the lines and angles, from which any person may at any time satisfy himself of the truth of the computations that have been made and are herein described. These measures I have here immediately subjoined, before I proceed to describe the computations made from them.
A Synopsis
A Synopsis of the horizontal and vertical angles that were observed at the principal points in making the survey about Schehallien.
In the first column are contained the names of the horizontal angles, the measure of which, in degrees and minutes, are in the second column, and the vertical angles are in the third column, in which it is to be observed, that the letter denoting the object is placed before the degrees and minutes, and E or D after them, to shew that they are in elevation or depression respectively. The mark * placed to the measure of any angle denotes that it is the mean of the two observations made with the instrument turned different ways; namely, after the first observation, reversing it to make the second. Also the mean height of the theodolite is put down to each station. In the vertical angles, the bottom of the object is understood, unless where the top is mentioned below, and sometimes the height of the pole is added in feet and inches.
| At A | Theodolite = 4 ft. 10 in. | At F | Theodolite = 4 ft. 8½ in. |
|------|--------------------------|------|--------------------------|
| DAB | 31° 15' | GFW | 17° 27' |
| DAN | 77° 30' N 12° 22E | GFK | 76° 7¼' K 9° 50' E |
| DAO | 102° 36½° O 0° 58E | GFN | 103° 57¼' N 17° 42½E |
| DAR | 134° 31½' | GFD | 172° 54' D 3° 37' E |
| NAC | 83° 13½' N 12° 20E | HFG | 10° 9' G 1° 36' D |
| NAO | 25° 4' | HFP | 60° 41' P 5° 9' E |
| OAS | 27° 24' S 5° 25D | HFK | 65° 58' K 9° 41' E |
| OAR | 31° 55' R 6° 31D | HFN | 93° 49' N 17° 35½E |
| At B | | At G | Theodolite = 4 ft. 6½ in. |
|------|---------------------------|------|--------------------------|
| DBN | 8° 1' | DGF | 6° 8' |
| DEO | 101° 41' | NGD | 59° 41' |
| DBA | 139° 59' | NGF | 65° 49' |
| | | NGH | 101° 9' |
| | | HGF | 166° 57' |
| | | HGD | 160° 49' |
| | | HGX | 71° 11½' |
| | | HGW | 4° 32' |
| | | PGF | 95° 56' F 1° 15' E |
| | | PGN | 30° 6½' N 16° 54' E top of Cairn. |
| | | PGK | 0° 9¼' K 10° 15' E |
| | | PGL | 62° 57' L 0° 32' D |
| | | PGW | 63° 20' W 0° 54' D |
| | | PGH | 71° 2' H 3° 0' D |
| | | | P 5° 46' E top of ball. |
At
### The mean Density of the Earth.
| At H | Theodolite = 4 ft. 5½ in. |
|------|--------------------------|
| GHF | 2° 54' |
| GHD | 13° 2' |
| NHG | 49° 44' |
| NHK | 29° 17' |
| NHW | 107° 43' |
| NHW | 107° 41' |
| KHD | 65° 59' |
| KHG | 79° 1' |
| KHW | 78° 25' |
| WHP | 96° 14' |
| WHP | 149° 2' |
| WHG | 157° 25½' |
| At W | Theodolite = 4 ft. 5½ in. |
|------|--------------------------|
| LWK | 107° 27½' |
| LWP | 130° 40' |
| LWP | 130° 41' |
| LWN | 133° 48½' |
| LWF | 175° 2' |
| LWQ | 4° 41' |
| LWG | 178° 21' |
| LWH | 193° 13' |
| At L | Theodolite = 4 ft. 9 in. |
|------|--------------------------|
| GLP | 37° 1' |
| GLY | 139° 32½' |
| WLG | 1° 18' |
| WLP | 38° 12' |
| At Y | Theodolite = 4 ft. 8 in. |
|------|--------------------------|
| TYK | 69° 23¼' |
| TYN | 76° 56½' |
| TYL | 124° 47' |
| TYG | 103° 24¼' |
| TYZ | 19° 21¼' |
| TYV | 13° 29½' |
| VYT | 13° 28' |
| VYZ | 32° 52' |
| VYE | 81° 49½' |
| VYK | 82° 53' |
| VYN | 90° 24' |
| VYG | 116° 52' |
| VYH | 126° 57' |
| VYW | 130° 33' |
| VYL | 138° 14' |
| At N | Theodolite = 4 ft. 5½ in. |
|------|--------------------------|
| WLN | 38° 58' |
| WLK | 60° 33' |
| WLT | 115° 38' |
| WLV | 126° 58½' |
| WLY | 140° 50½' |
| WLF | 3° 58½' |
| WLE | 62° 12' |
| WLT | 38° 4½' |
| At K | Theodolite = 4 ft. 8 in. |
|------|--------------------------|
| K 10° 29' E |
| N 9° 37½' E |
| L 2° 35' E |
| G 1° 36½' E |
| Z 3° 3½' E |
| V 3° 26½' E |
| T 2° 4½' E |
| Z 3° 3½' E |
| E 10° 33' E |
| K 10° 29' E |
| N 9° 30½' E |
| G 1° 35½' E |
| H 1° 0' E |
| W 2° 2' E |
| L 2° 33½' E |
At
| At v | Theodolite = 4 ft. 11 in. |
|------|--------------------------|
| TVZ | 17 25½ * |
| | 17 25 * |
| TVU | 34 24½ * |
| TVF | 70 43 * |
| TVK | 71 23½ * |
| | 71 24 * |
| TVW | 112 32½ * |
| TVL | 119 57 * |
| TVY | 147 5½ * |
| | 147 50 * |
| UZY | 117 21 * |
| UZV | 134 4½ * |
| UZT | 144 27½ * |
| At u | Theodolite = 4 ft. 10 in. |
|------|--------------------------|
| ZUT | 18 20½ * |
| ZUV | 28 54½ * |
| ZUO | 124 42 * |
| ZUN | 125 9½ * |
| ZUX | 128 2½ * |
| ZUA | 157 55 * |
| ZUR | 160 37½ * |
| At t | Theodolite = 4 ft. 10 in. |
|------|--------------------------|
| YTV | 18 41½ * |
| YTL | 30 1 * |
| YTU | 116 18½ * |
| YTZ | 133 3½ * |
| VTY | 18 41 * |
| VTL | 48 43 * |
| VTE | 94 2 * |
| VTU | 135 1½ * |
| VTZ | 152 14½ * |
| At x | Theodolite = 4 ft. 7 in. |
|------|--------------------------|
| OXA | 40 59½ * |
| OXS | 53 20 * |
| OXZ | 139 22½ * |
| OXU | 174 59 * |
| At s | Theodolite = 4 ft. 9 in. |
|------|--------------------------|
| RSA | 18 27 |
| RSA | 18 28 |
| RSN | 72 51 |
| RSO | 90 58 |
| RSX | 171 1 |
| At z | Theodolite = 4 ft. 6½ in. |
|------|--------------------------|
| UZR | 10 53½ * |
| UZA | 16 5 * |
| UZX | 16 18 * |
| UZO | 33 18½ * |
| A 5 24½ E |
| A 5 20½ E |
| N 18 40 E |
| O 13 19½ E |
| X 0 55½ D |
| R 0 24½ E |
### The mean Density of the Earth.
#### At R
Theodolite = 4 ft. 6 in.
| KRN | 41 6½ |
|-----|-------|
| B''RA | 20 36½ * A 6 * 27½ E |
| B''RN | 84 32½ * N 19 * 17½ E |
| B''RO | 111 46½ * O 11 * 57½ E |
| B''Rα' | 100 14½ * α' 20 * 2 E |
| B''Rβ' | 121 27½ * β' 15 * 11½ E |
| B''RS | 177 39 |
| B''RZ | 186 53½ * Z O * 46 E |
| its sup. | 173 6½ * O 12 29 E |
#### At B''
Theodolite = 4 ft. 10 in.
| RB''S | 0 55½ |
|-------|-------|
| RB''β' | 36 8½ * β' 10 * 24 E |
| RB''O | 37 27 * O 7 * 35½ E |
| RB''α' | 50 49½ * α' 15 * 48 E |
| RB''N | 65 46½ * N 17 * 33 E |
| RB''A | 139 2½ * A 11 * 22½ E |
#### At o
Theodolite = 4 ft. 7 in.
| XOS | 35 39 * S 13 * 25½ D |
|-----|-----------------------|
| XOM | 42 12½ * M 13 * 12½ D |
| XOR | 58 49½ * R 12 * 4 D |
| XOB'' | 89 35 * B'' 7 * 39½ D |
| XOA | 115 43½ * A O * 59 D |
| XOY | 1 39 * U 4 * 4½ D |
#### At P
Theodolite = 4 ft. 10 in.
| NPK | 77 18½ |
|-----|--------|
| GPK | 179 35½ * G 5 39½ D |
| GPH | 47 45½ * H 7 * 32½ D |
| GPW | 69 0½ * W 5 * 37 D |
| GPL | 80 9½ * L 4 * 27½ D |
| GPF' | 20 54 * F' 5 * 6 D |
| GPm | 38 18½ * m 12 * 20 D |
| γ'Pγ' | 45 39 |
| γ'PK | 2 55½ |
* is in the meridian passing through o, and is a little South of the intersection of that meridian and the line as.
γ' from p bears 30° 41' 2½ E. of South.
δ' from p bears 14° 57' W. of South.
m is Mr. Mason's mark.
| At N | Theodolite = 4 ft. 11 in. | At K | Theodolite = 4 ft. 8 in. |
|------|--------------------------|------|--------------------------|
| KNO | 40° 5½' | NKO | 44° 51¾' |
| KNR | 74° 48½' | NKR | 64° 5½' |
| KNP | 50° 47½' | NKP | 51° 54½' |
| KNδ' | 9° 37½' | NKn | 0° 45° |
| KNγ' | 2° 11½' | NKF | 43° 32½' |
| KNA | 133° 53° * | NKG | 5° 40° * |
| KNB | 146° 15½° * | NKα | 55° 12° |
| KNC | 178° 30½° * | NKH | 81° 28° |
| KND | 172° 51° * | NKH | 81° 27½° * |
| KNη | 144° 37½° * | NKW | 97° 18° * |
| KNG | 98° 23° * | NKL | 109° 19½° * |
| KNH | 69° 13½° | NKL | 109° 18½° * |
| KNm | 64° 27° * | NKY | 153° 40° * |
| KNW | 56° 22° * | NKV | 174° 20° * |
| KNL | 49° 7° * | NKE | 156° 18° |
| KNY | 18° 50° * | NKα' | 11° 36° * |
| KNγ' | 2° 15° * | NKβ' | 6° 2½° * |
| KNE | 1° 15½° * | NKα' | 11° 36° * |
| KNδ' | 9° 37½° * | NKβ' | 6° 2½° * |
| KNU | 41° 4½° * | NKα' | 11° 36° * |
| KNS | 60° 44° * | NKα' | 11° 36° * |
| KNR | 74° 48½° * | NKα' | 11° 36° * |
| KNβ' | 104° 30° * | NKα' | 11° 36° * |
At
| At α | Theodolite = 4 ft. 4 in. | At n | Theodolite = 4 ft. 9 in. |
|------|--------------------------|------|--------------------------|
| γαK | 106 0 * K 10 * 1 2E | Knα | 108 59 1/2 * K 6 * 40 1/2D |
| γαG | 101 3 * G 9 * 37 1/2E | Knβ | 128 35 1/2 * β 13 * 7 1/2D |
| γαN | 89 41 * N 13 * 45 1/2E | KnF' | 128 55 1/2 * F' 14 * 10 1/2D |
| γαn | 89 25 * n 13 * 49 1/2E | Knγ | 135 40 1/2 * γ 12 * 29 D |
| γαF' | 47 38 * F' 8 * 6 1/2E | KnD | 173 55 D 10 * 45 D |
| | β 0 19 D | KnN | 38 28 1/2 * N 1 42 D |
| | γ o o top Pole 3 ft. 2 in.| KnB''| 102 54 1/2 * B'' 17 31 D |
| | | DNA | 53 49 * A 12 * 21 1/2D |
| | | DnG | 75 24 * G 17 * 3 1/2D |
| | | DnP''| 78 44 1/2 * P'' 20 * 2 1/2D |
At β between α and γ the ends of the N. base.
| αβk | 174 51 k 0 41 1/2E | DnH | 104 35 H 14 * 42 1/2D |
| αβt' | 147 37 1/2 t' 1 * 25 1/2E| DnL | 124 34 L 10 56 D |
| αβF' | 71 49 * F' 10 * 22 1/2E | DnK | 173 55 α 13 * 51 1/2D |
| αβD | 108 14 * D 9 * 17 1/2E | NnG | 132 59 α 13 * 54 D |
| αβn | 70 58 1/2 * n 13 * 5 1/2E| | |
| γβD | 71 57 1/2 * D 9 * 17 1/2E| | |
| γβF' | 108 9 1/2 * F' 10 * 22 1/2E| | |
At γ the Western end of the North base.
| αγF' | 50 6 1/2 * F' 8 * 23 1/2E| NEA | 10 5 1/2 * A 6 * 22 1/2E |
| αγn | 63 53 1/2 * n 12 * 27 1/2E| NEM' | 4 58 1/2 * M' 4 * 45 1/2E |
| αγD | 97 5 1/2 * D 9 * 41 E | NED | 4 51 1/2 * D 6 * 41 1/2E |
| | α o * 2 D top Pole 3 ft. 2 in.| NEK | 22 14 1/2 * K 2 * 11 1/2E |
| | | NEH | 78 40 1/2 * H 11 * 47 1/2D top Pole 6 ft. 5 in. |
| | | NEW | 94 17 * W 11 * 16 1/2D top Pole 7 ft. |
| | | NEL | 106 21 1/2 * L 10 * 17 D top Pole 6 ft. 8 in. |
Vol. LXVIII. 4 R NEY
| NEY | 151 16 * Y 10 * 37 D |
| NEV | 172 20 * V 9 * 15 D |
| NET | 172 22 1/2 * T 9 * 41 1/2 D top |
| NEZ | 144 46 * Z 9 * 18 1/2 D |
| NEU | 114 20 1/2 * U 10 * 4 1/2 D |
| a'Ea' | 68 35 1/2 * a' 6 * 21 E |
| a'Eδ | 73 50 1/2 * δ 6 * 37 1/2 E |
| a'EN | 78 39 1/2 * N 6 * 33 E |
| a'Eb' | 97 17 * b' 9 * 5 1/2 D |
| a' | 8 * 27 D |
At M', the meridian mark on the top of the hill South off.
| KM'E | 3 28 1/2 * E 4 * 56 1/2 D |
| KM'γ' | 6 37 1/2 * γ' 6 * 3 1/2 D top |
| KM'L | 53 27 1/2 * L 10 * 19 D top Pole 6 ft. 8 in. |
| KM'W | 63 12 1/2 * W 11 * 41 D top Pole 7 ft. |
| KM'H | 78 4 1/2 * H 12 * 56 1/2 D top Pole 6 ft. 5 in. |
| KM'm | 86 33 1/2 * m 22 * 12 D |
| KM'p | 87 54 * p 22 * 10 D |
| KM'G | 108 49 * G 12 * 33 D top Pole 4 ft. 4 in. |
| KM'F | 118 3 1/2 * F 12 * 25 D top Pole 4 ft. 2 in. |
| KM'δ | 176 1 1/2 * δ' 12 * 8 E |
| K | 5 5 D |
At t, the center of the transit instrument near the N. observ.
| M''tP' | 32 24 * P' 9 * 18 1/2 D |
| M''tG | 32 57 1/2 * G 5 * 58 1/2 D |
| M''tm | 5 14 * m 12 * 32 1/2 D |
| M''th | 14 28 * H 7 * 44 1/2 D |
| M''tw | 35 23 1/2 * W 5 * 47 1/2 D |
| M''tl | 46 22 * L 4 * 35 1/2 D |
| mtp | 68 13 * p 12 * 50 1/2 E |
| mtM' | 68 15 * M' 22 * 5 E |
| mtG | 144 17 * G 5 * 57 1/2 D |
| mth | 96 52 * H 7 * 43 1/2 D |
| mtw | 75 56 * W 5 * 47 1/2 D |
| mtl | 64 58 * L 4 * 37 1/2 D |
| mtp | 54 4 1/2 * P 17 * 42 1/2 D top |
M'' bears 1' 8" W. of North.
M' bears South.
P' in a line with K and p'.
p, a pole immediately above or South of the transit instrument.
At p, Theodolite = 4 ft. 5 in.
| GpM' | 147 5 * M' 22 * 8 1/2 E |
| Gpδ | 133 6 1/2 * δ' 24 * 18 E |
| GpF | 26 43 * F 5 26 D |
| GpP' | 0 34 1/2 * P' 9 * 23 1/2 D |
| GpM'' | 32 47 1/2 * M'' 13 * 25 1/2 D |
| Gpt | 32 48 * t 19 17 D |
| Gpm | 38 0 1/2 * m 12 * 34 1/2 D |
| GpH | 47 11 * H 7 * 49 1/2 D |
GpW
| GpW | 68 3½ * | W 5*51½D |
| GPP | 76 55½ * | P 13*49½D |
| GPL | 79 0½ * | L 4*39½D |
| Gpm' | 121 24 * | m' 4*50½D |
| GPE | 179 37 * | E 17*39½E |
*m' is a pole a little above or South of P.
At m'
| Gm'M' | 145 4½ * | M'22*8½E |
| Gm'N | 131 11 * | N 24*12½E |
| Gm'P | 57 53½ * | P 2*47 D |
| Gm'T | 34 43 | T 10 15 D |
| Gm'F | 13 1 * | F 5*22½D |
| Gm'P' | 0 29½ * | P'19*19 D |
| Gm'M'' | 32 29½ * | M''13*23½D |
| Gm'P | 37 6 * | P 18*16½D |
| Gm'm | 37 49½ * | m 12*36½D |
| Gm'H | 47 10 * | H 7*49 D |
| Gm'W | 68 11 * | W 5*49½D |
| Gm'L | 79 17½ * | L 4*40½D |
| Gm'E | 177 53 * | E 17*54 E |
At m
| pmK | 15 57 | K 15 O E |
| pma | 37 11 | P 12 27 E |
At a'
Theodolite = 4 ft. 11 in.
| Ka'N | 149 3 * | K 6*43½D |
| Ka'B' | 131 3 * | B'15*51½D |
| Ka'R | 102 7½ * | R 20*9 D |
At a'
Theodolite = 4 ft. 9 in.
| Ka'O | 73 39 * | O 33*52 D |
| Ka'B' | 28 4½ * | B'15*56½D top |
| Ka'γ' | 4 53½ * | γ' 7*35½ top |
| Ka'δ' | 65 24 * | δ' 3 41½ top |
| Pole 17 ft. 4 in. |
At γ'
Theodolite = 4 ft. 4 in.
| Ny'P | 56 55½ * | P 18*3½D |
| Ny't | 55 22½ * | t 17 49½D |
| Ny'n | 0 50½ * | n 7 15 E |
| Ny'δ' | 8 17½ * | δ' 7*59½E |
| Ny'a' | 14 17 * | a' 7*31 E |
| Ny'K | 151 54½ * | K 1*34½E |
| N 7*23½E |
4 R 2
| At δ' | Theodolite = 4 ft. 4 in. | At k | Theodolite = 4 ft. 10½ in. |
|-------|--------------------------|------|--------------------------|
| Kδ'M' | 2 53 * M' 12 * 36½ D | t'kF' | 4 42 F 5 11 E |
| Kδ'γ' | 4 29 * γ' 8 * 8½ D | t'ka | 34 47 α 0.27 D |
| Kδ'm' | 73 20½ * m' 24 * 9½ D top | t'kβ | 37 13 β 0 52 D |
| Kδ'p' | 74 35½ * p 24 * 16½ D top | t' | 0 26 E |
| At a | Theodolite = 4 ft. 8½ in. |
|------|--------------------------|
| baK | 113 57½ K 14 45 E |
| baN | 155 24 Top of the cairn. |
| baw | 80 40 |
| baL | 59 48 |
| At b | Theodolite = 4 ft. 6 in. |
|------|--------------------------|
| abK | 43 35 K 13 32 E |
| abN | 18 49 N 11 26 |
| abL | 5 39 |
| At d | Theodolite = 4 ft. 9 in. |
|------|--------------------------|
| cdN | 14 11 N 11 33 E |
| cdG | 34 53 |
| cdH | 63 27 H 6 36 D |
| cdL | 111 12 L 6 42 D |
| At c | Theodolite = 4 ft. 8 in. |
|------|--------------------------|
| dcE | 49 34 * L 7 * 21½ D |
| dcH | 97 9 * H 9 * 17½ D |
| dcG | 136 9½ * G 4 * 58 D |
| dcK | 118 55½ * K 14 * 35½ E |
| dcF | 144 20 F 4 24 D |
At
### The Mean Density of the Earth
| At \(a'\) | Theodolite = 4 ft. 10 in. | At \(c'\) | Theodolite = 4 ft. 8\(\frac{1}{2}\) in. |
|-----------|--------------------------|-----------|----------------------------------------|
| \(Ea'a'\) | 102° 4' 8" * | \(a'\) | 7° 56' E |
| \(Ea'b'\) | 97° 5' 4" * | \(b'\) | 6° 4' D |
| \(Ea'b'\) | 65° 58' 2" * | \(c'\) | 11° 20' 2" D |
| \(Ea'd'\) | 108° 2' 1" * | \(d'\) | 2° 34' 2" E |
| | E 7° 19' E | | |
| At \(b'\) | Theodolite = 4 ft. 9\(\frac{1}{2}\) in. |
|-----------|----------------------------------------|
| \(c'b'a'\) | 36° 41' 4" * |
| \(c'b'E\) | 53° 26' 1" * |
| At \(d'\) | Theodolite = 4 ft. 5 in. |
|-----------|--------------------------|
| \(a'd'a'\) | 169° 14' 3" * |
| \(a'\) | 12° 35' 1" E |
| \(a'\) | 2° 48' D |
Several other angles and bearing of objects were taken, which, being of no use in computing the attraction of the hill, are here omitted.
The foregoing tables, containing all the angles collected together which were observed at the same point, include all the horizontal angles that were at different times taken for ascertaining the relative places of the principal points and objects on a horizontal plane. The numerous other angles used, in finding the sections of the ground, are given hereafter, with their computed results annexed to them.
We now proceed to speak of the two principal bases which were accurately measured, as foundations on which every thing else must depend; and first,
Of the measure of the base RB" in Glenmore, the valley on the South of Schehallien, taken the 16th, &c. of Sept. 1774.
Here A and B are the names of the two measuring rods, which were laid down alternately in the order as expressed in the following table of measures. The lengths of these rods, by the brass standard, when the thermometer was at $62\frac{3}{4}$, were thus, viz.
$$\begin{align*}
A &= 20 \text{ feet } 1'255 \text{ inch.} = 20.10458 \\
B &= 20 \text{ feet } 1'323 \text{ inch.} = 20.11025
\end{align*}$$
feet.
The numbers following each rod, with the sign + interposed, are inches and decimal parts; and they denote the distance beyond the end of each rod to the beginning of the next following rod; and, therefore, the sum of all these numbers must be added to the sum of the lengths of the rods themselves for the total of the measures. Also, as the first rod began at 2 feet 8 inches from the point R, this number is to be added to the total last mentioned, to give the measure of the whole base from R to B".
$A+8.29$
The mean Density of the Earth.
| A +8·29 | B +3·45 | B +7·22 | A +3·34 | A +4·47 | A +3·69 |
|---------|---------|---------|---------|---------|---------|
| B +2·53 | A +3·80 | A +1·91 | B +3·65 | B +3·75 | B +4·07 |
| A +6·11 | B +6·64 | B +4·46 | A +6·96 | A +4·74 | A +2·75 |
| B +6·66 | A +7·76 | A +1·95 | B +3·07 | B +3·06 | B +4·65 |
| A +2·79 | B +3·28 | B +2·26 | A +3·55 | A +2·58 | A +4·07 |
| B +1·20 | A +4·87 | A +4·54 | B +2·93 | B +4·15 | B +3·23 |
| A +2·07 | B +6·18 | B +4·48 | A +5·53 | A +3·16 | A +4·16 |
| B +4·80 | A +8·70 | A +3·14 | B +5·33 | B +4·64 | B +5·73 |
| A +0·00 | A +7·87 | B +3·38 | A +4·38 | A +3·26 | A +4·12 |
| B +1·78 | B +4·75 | A +5·00 | B +5·67 | B +4·18 | B +4·91 |
| A +3·29 | A +6·56 | B +4·85 | A +5·12 | A +4·04 | A +3·18 |
| B +2·85 | B +5·24 | A +6·12 | B +1·06 | B +2·92 | B +3·91 |
| A +6·39 | A +7·90 | B +3·44 | A +5·96 | A +3·10 | A +5·28 |
| B +4·86 | B +6·32 | A +6·18 | B +2·47 | B +5·11 | B +2·90 |
| A +6·08 | A +6·92 | B +4·19 | A +3·84 | A +4·61 | A +4·39* |
| B +8·58 | B +7·28 | A +4·51 | B +5·57 | B +3·34 | B +4·37 |
| A +9·07 | A +6·34 | B +3·04 | A +2·63 | A +2·57 | A +3·29 |
| B +1·53 | B +8·93 | A +4·37 | B +7·41 | B +5·80 | B +2·12 |
| A +2·28 | A +5·39 | B +2·96 | A +3·11 | A +3·37 | A +2·95 |
| B +7·47 | B +5·20 | A +2·47 | B +1·74 | B +2·58 | B +3·30 |
| A +2·40 | A +3·54 | B +3·90 | A +2·07 | A +2·24 | A +2·82 |
| B +5·42 | B +1·26 | A +5·78 | B +4·33 | B +3·48 | B +3·97 |
| A +7·42 | A +3·20 | B +3·97 | A +5·93 | A +2·95 | A +1·37 |
| B +8·14 | B +5·34 | A +4·87 | B +6·36 | B +2·88 | B +0·00 |
| A +8·77 | A +3·74 | B +4·83 |
The sum of all these is $74A + 73B + 669\cdot28$ inches,
or $74A + 73B + 55\cdot773$ feet, including the 2 feet 8 inches at the beginning of the measurement.
Now \(74A\) is = \(1487.73892\)
\(73B\) is = \(1468.04825\)
The odd parts \(\frac{55}{773}\)
Sum \(\frac{3011.56017}{\text{the base unreduced.}}\)
But a reduction of this must be here made according to the state of the thermometer, and for the wearing of the brass 5 feet standard (see Phil. Transf. vol. LVIII. for the year 1768, p. 313, &c.). Now the difference between \(62^\circ\) and \(62\frac{3}{4}\) being \(\frac{3}{4}\), therefore \(3011.56 \times \frac{232}{180000} \times \frac{3}{4} = 3011.56 \times \frac{29}{270000} \times \frac{3}{4} = 0.024\) feet, is the small correction on account of the thermometer, and which being added makes the number become \(3011.584\) for the length of the base as reduced to the state of \(62^\circ\) of Fahrenheit's thermometer. But the brass rod had been \(\frac{1}{1000}\)th of an inch shortened by wearing, and it was originally \(\frac{1}{1000}\)th of an inch shorter than the Royal Society's brass standard yard, so that it is now \(\frac{1}{500}\)dth inch shorter than that standard in the length of 3 feet, or \(\frac{1}{1800}\)th part of the whole; therefore subtracting the \(\frac{1}{1800}\)th part, or \(0.167\) from the above quantity, there remains \(3011.417\) feet for the corrected measure of this base, or the true length of the line \(RB''\).
The above measures, as far as to that marked * inclusively, together with 10 feet \(10\frac{1}{2}\) inches more, reach
to a place to which they had before measured with the tape line, and by it found to be $2844.8$ feet; while the measure of the same by the rods is found to be $2839.3$ feet. The difference is $5\frac{1}{2}$ feet, a small part of which might be owing to the unstable state of the wooden stands used in the first quarter of the base; but the greater part of this difference is more likely to be owing to the uncertain way of measuring with a tape, which, to say nothing of the ground not being quite level, is liable to be stretched more or less in length with different degrees of tension, and to be variously warped in length by moisture.
Of the measurement of the base $\alpha\beta\gamma$ in Rannoch, to the North-west of the hill of Schehallien.
1. One part of this base was measured twice over in different ways. The part $\alpha\beta$ was carefully measured on the 8th of October 1774 with a chain, and found to be 63 chains and $40\frac{1}{2}$ links, or $63.405$ chains in length.
Now on the 24th of the same month the chain was measured by means of the five-feet brass standard, when the thermometer was at $38^\circ\frac{1}{2}$, and the length found to be $65.94542$ feet. Hence then $65.94542 \times 63.405 = 4181.269$ is the length of all the chains, to which, add-
Mr. Hutton's Calculations to ascertain
ing 1.764 the breadth of the 63 iron pins, the sum is
4183.033 for the length of \( \alpha \beta \) uncorrected.
But \( 62 - 38\frac{1}{2} = 23\frac{1}{2} \), therefore \( -23\frac{1}{2} \times \frac{29}{2700000} \times 4183 \)
\( = -1.056 \) is the reduction on account of the state of the
thermometer, which being applied with its proper sign,
there results 4181.977; and from this last number de-
ducting again \( \frac{1}{18000} \)th part or .232, on account of the
wearing of the brass standard, there then remains
4181.745 feet for the length of the part \( \alpha \beta \) of the base
in Rannoch, as measured by the chain.
But as the chain was measured not at the same time
with the base, but between two and three weeks later,
when the air was probably cooler, the reduction above
made for the state of the thermometer is perhaps some-
thing too great, and we may safely conclude \( \alpha \beta \) to be
equal 4182 feet as measured by the chain.
2. The whole base \( \alpha \beta \gamma \) was next, on the 10th, 11th,
and 12th of October, very carefully measured by the
twenty-feet measuring rods. The rods at that time mea-
sured thus, \( \begin{cases} A = 20 \text{ ft.} + 1.306 \text{ inch.} = 20.108\frac{5}{6} \\ B = 20 \text{ ft.} + 1.354 \text{ inch.} = 20.112\frac{5}{6} \end{cases} \) feet;
the thermometer being then at 40°. The number of
rods and the additional parts were as follows.
A + 4.49
| A +4'49 | A +3'99 | B +1'73 | B +3'22 | B +2'43 | B +1'87 |
|---------|---------|---------|---------|---------|---------|
| B +3'29 | B +3'13 | A +2'41 | A +3'16 | A +1'97 | A +2'38 |
| A +6'57 | A +3'55 | B +2'99 | B +2'07 | B +1'74 | B +2'67 |
| B +3'62 | B +4'19 | A +2'62 | A +2'22 | A +2'84 | A +2'07 |
| A +3'84 | A +4'06 | B +2'18 | B +4'83 | B +3'65 | B +1'84 |
| B +3'52 | B +3'94 | A +2'72 | A +2'39 | A +1'62 | A +4'03 |
| A +4'50 | A +3'64 | B +3'02 | B +2'68 | B +1'27 | B +1'72 |
| B +3'62 | B +3'23 | A +2'46 | A +2'09 | A +2'38 | A +2'14 |
| A +4'88 | A +3'76 | B +3'68 | B +1'97 | B +2'36 | B +1'80 |
| B +2'74 | B +2'56 | A +2'62 | A +1'48 | A +2'57 | A +1'27 |
| A +3'24 | A +3'38 | B +2'72 | B +3'20 | B +2'07 | B +1'88 |
| B +4'30 | B +3'29 | A +3'33 | A +2'62 | A +2'48 | A +2'36 |
| A +3'50 | A +3'65 | B +2'93 | B +2'37 | B +2'31 | B +2'04 |
| B +3'26 | B +3'51 | A +3'14 | A +2'47 | A +2'28 | A +2'37 |
| A +2'96 | A +3'88 | B +2'93 | B +3'48 | B +3'96 | B +1'77 |
| B +3'32 | B +2'29 | A +2'40 | A +3'16 | A +4'87 | A +1'66 |
| A +5'93 | A +3'13 | B +2'06 | B +3'50 | B +2'61 | B +2'26 |
| B +5'00 | B +3'71 | A +3'13 | A +2'33 | A +2'22 | A +1'97 |
| A +3'43 | A +3'13 | B +2'57 | B +2'37 | B +1'63 | B +1'77 |
| B +3'87 | B +3'13 | A +2'85 | A +2'68 | A +1'87 | A +2'63 |
| A +6'37 | A +5'43 | B +2'87 | B +2'68 | B +2'41 | B +3'18 |
| B +3'48 | B +3'08 | A +4'12 | A +2'70 | A +2'62 | A +2'09 |
| A +4'86 | A +3'57 | B +2'20 | B +2'68 | B +2'09 | B +1'74 |
| B +4'87 | B +5'68 | A +2'68 | A +2'05 | A +2'27 | A +2'74 |
| A +3'08 | A +3'89 | B +2'17 | B +3'09 | B +3'02 | B +4'49 |
| B +3'67 | B +2'78 | A +2'13 | A +2'50 | A +2'41 | A +2'60 |
| A +3'28 | A +3'27 | B +2'17 | B +2'52 | B +2'53 | B +2'45 |
| B +3'43 | B +1'84 | A +3'12 | A +2'62 | A +1'84 | A +1'41 |
| A +4'82 | A +0'00 | B +2'70 | B +2'43 | B +2'68 | B +2'67 |
| B +4'46 | A +3'14 | A +3'17 | A +3'01 | A +2'37 | A +1'98 |
4 S 2
B +2'96
| B +2·96 | A +1·66 | B +2·77 | B +2·37 | B +1·86 | B +2·49 |
|---------|---------|---------|---------|---------|---------|
| A +2·27 | B +1·53 | A +2·09 | A +2·86 | A +2·33 | A +2·11 |
| B +2·60 | A +1·80 | B +2·14 | B +2·42 | B +2·65 | B +3·03 |
| A +3·08 | B +3·53 | A +2·45 | A +2·16 | A +2·14 | A +2·67 |
| B +2·23 | A +2·45 | B +2·98 | B +2·21 | B +2·35 | B +2·43 |
| A +4·70 | B +0·00* | A +2·40 | A +2·75 | A +2·43 | A +1·93 |
| B +2·28 | A +2·36 | B +2·50 | B +2·03 | B +2·59 | B +2·75 |
| A +1·79 | B +2·75 | A +2·82 | A +2·73 | A +2·48 | A +1·99 |
| B +1·83 | A +1·77 | B +2·37 | B +2·14 | B +2·48 | B +2·17 |
| A +2·74 | B +1·55 | A +2·76 | A +1·57 | A +2·91 | A +1·83 |
| B +2·48 | A +1·97 | B +2·91 | B +1·77 | B +2·47 | B +1·93 |
| A +2·23 | B +2·04 | A +2·58 | A +2·21 | A +1·80 | A +1·65 |
| B +1·66 | A +2·56 | B +2·34 | B +2·11 | B +1·99 | B +1·04 |
| A +3·08 | B +2·15 | A +2·86 | A +2·99 | A +2·81 | A +1·96 |
| B +2·31 | A +2·26 | B +2·67 | B +2·06 | B +2·44 | B +2·77 |
| A +2·20 | B +2·37 | A +2·19 | A +2·36 | A +2·07 | A +2·25 |
| B +3·06 | A +1·94 | B +2·37 | B +1·77 | B +2·44 | B +1·79 |
| A +2·36 | B +1·97 | A +2·93 | A +1·66 | A +2·11 | A +0·00 |
| B +2·24 | A +1·94 |
Of the foregoing measures, the sum of all from the beginning to that marked * inclusively, together with 13 feet 2 inches more, brings us to the point β before measured to by the chain. Now to this place, by adding together the measures, there are found to be 103 A and 102 B, and the sum of the parts is 586·71 inches.
Then
Then $103A = 103 \times 20'108\frac{5}{6} = 2071.210$
$102B = 102 \times 20'112\frac{5}{6} = 2051.509$
$586.71$ inches $= 48.893$
$13$ ft. 2 inch. $= 13.167$
Hence $\alpha\beta$ (unreduced) is $4184.779$
But since $62 - 40 = 22$, therefore the reduction for the state of the air is $-22 \times \frac{29}{2700000} \times 4185 = -0.989$, which being applied to the above sum, there remains $4183.79$ as corresponding to the state of $62^\circ$ of the thermometer.
From this last number deduct its $\frac{1}{18000}$th part, viz. $0.232$, and there results $4183.558$ for the correct length of the part $\alpha\beta$ as determined by this very accurate method; which is but about a foot and a half more than what it was found to be by the less accurate measure by the chain, which is a nearer approach to an equality than could well be expected.
To determine now the whole length of the base $\alpha\gamma$; by taking the whole sums there are found to be $146A$ with $144B$ and $779.78$ inches of the odd parts.
Then $146A = 146 \times 20'108\frac{5}{6} = 2935.890$
$144B = 144 \times 20'112\frac{5}{6} = 2896.248$
$779.78$ inches $= 64.982$
The sum or $\alpha\gamma$ (unreduced) is $5897.120$.
The correction for the thermometer is $-22 \times \frac{29}{270000} \times 5897 = -1.394$, which being applied to the number above, there results $5895.726$; and this again being diminished by its $\frac{1}{18000}$th part, or $0.327$, there remains $5895.399$ feet, for the correct measure of the base $\alpha\gamma$ in the vale of Rannoch.
There is no occasion here to explain the manner of measuring these two bases by the twenty-foot rods, as that has been very circumstantially done in vol. LXV. of the Phil. Trans. for the year 1775, by the rev. Dr. Maske Lyne, the learned and accurate conductor of this very important experiment.
The following shorter lines were also measured as they happened to be wanted in different parts of the survey.
| Feet | Inch |
|------|------|
| $\alpha'\delta'$ | 269 4 |
| $nn$ | 93 6 |
| $ke$ | 94 10 |
| $ke$ | 240 10 |
| $ac$ | 9 9 |
| $an$ | 7 10 |
| $cn$ | 1 11 |
| $ma$ | 70 11 |
| $mt$ | 68 3 |
| $mp$ | 63 4 |
| $pt$ | 27 2 |
nearly horizontal.
not horizontal.
The other measures that were taken for determining the sections will be delivered afterwards, when the results or computed altitudes have been obtained, in order to be placed opposite to their correspondent angles.
Having now obtained, to a great degree of accuracy, the measured lengths of two lines which were to serve as bases for all the future calculations, the next consideration was how to make the properest use of them. Every other line or distance, drawn or conceived to be drawn, must be calculated from them by the help of the angles observed either at their extremities, or at all the other points and stations in the survey and plan. As these two bases are situated in the low parts of the country, from whence but a very few of the other principal stations are visible, one method evidently is to compute immediately from these bases such of the great lines in the survey whose extremities are visible from them; and then from these calculated lines to compute others next to them, and so on quite around and within the whole figure. In this manner several values of each line will arise, both from the double computations by the two measured bases, and from the various sets of triangles which can be formed from the very numerous horizontal angles which were observed at the several stations. But in this mode of computation, after
great labour and pains, I had frequently the mortification to find that the several values of the same lines would differ so greatly one from another, that I was often very doubtful whether I could rely on any of them, or even on the mean among them all. These differences arose from the small errors in the observed angles, which in some degree are unavoidable; and indeed they were so small, that the sum of the angles of the several triangles which were used in the calculation seldom differed by more than a minute or two from $180^\circ$. But in a long connected chain of triangles, dependant on one another, the effects of such small errors at length become too great to be tolerated in a computation requiring much accuracy. Another method is, first to compute from both bases the length of the line $KN$ extended along the ridge of the hill from East to West, and from it, as a secondary base, compute all the other lines in the plan. This method admits of much more accuracy than the former, supposing this secondary base to be truly assigned; because that, from the elevated and central situation of this line, all or most of the other points in the survey are visible from one or both of its extremities, by which it happens that the other lines are mostly determinable from it alone, without so close a connection with one another as in the other method of computation. By both
both of these methods then, and by all the triangles furnished by each of them, I computed all the principal lines in the plan, and either took a mean among the several values of each, or else selected out of them such one as from various circumstances I judged it safest to rely upon, as nearest the truth. The trigonometrical computations were always accurately made, and generally repeated by logarithms, and the result of every proportion determined to two or three places of decimals. I shall here abstract the mean or corrected values of some of the principal lines or horizontal distances so computed, as well as the secondary base $KN$ from the Eastern to the Western cairn.
The mean among a great number of ways of computation from the South base gives the horizontal distance from $K$ to $N = 4052.2$, and the mean of all the results from the North base $\alpha\beta\gamma$ gives $KN = 4058.9$, and the mean between these two gives $4055.5$ for the mean distance of $K$ and $N$. And this value of $KN$ was used in computing most of the other lines, whose mean results are as here follows.
\[ \alpha \gamma = 5895.4 \text{ the Northern base in Rannoch.} \]
\[ RB'' = 3011.4 \text{ the Southern base in Glenmore.} \]
\[ NK = 4055.5 \text{ the distance of the two cairns.} \]
| RA = 5670 | NR = 5545 | KR = 5952 | OR = 3582 |
|-----------|----------|----------|----------|
| AB = 1489 | NE'' = 6053 | KF = 8227 | OB'' = 5466 |
| BC = 4506 | NA = 5941 | KG = 8036 | OA = 6769 |
| CD = 775 | NB = 6573 | KH = 7748 | OS = 3271 |
| DF = 7388 | NC = 7797 | KW = 7603 | OX = 4079 |
| FG = 1166 | ND = 7657 | KL = 8335 | OU = 6061 |
| GH = 4068 | NF = 5980 | KY = 10008 | OZ = 9073 |
| HW = 2118 | NG = 6370 | KV = 10215 | OM = 3317 |
| WL = 1816 | NH = 8195 | KO = 2615 |
| LY = 7085 | NW = 9059 | KP = 3221 |
| YV = 3636 | NL = 10405 | KA = 13710 | MS = 381 |
| VT = 2645 | NY = 13752 | KB = 15404 | TE = 1335 |
| TZ = 4393 | NS = 5795 | KM' = 1817 | ZE = 3719 |
| ZU = 4132 | NO = 2875 | KD' = 2528 | FD' = 6430 |
| UX = 1984 | NP = 3271 | KA = 3326 | FF' = 3934 |
| XS = 2378 | NA = 11876 | KB = 4409 | F'T' = 4098 |
| SR = 1410 | NA = 5899 |
| NB = 7614 |
| NB = 3381 | PG = 4815 | ab = 1843 |
| ND' = 1585 | PH = 5196 | cd = 1750 |
From the three first lines, or bases, and the horizontal angles observed at the several stations, a very large and accurate plan of the whole survey was constructed, forming a map of four feet long by four feet broad, which was verified in every part by the measures of the computed
puted lines, both those above-written and others, and they were generally found to agree very exactly, according to the scale by which the plan was constructed. The use of this large map was to receive and admit of the distinct and accurate exhibition of the figures in their true places, expressing the number of feet in elevation or depression with respect to each observatory of every point and section of the ground whose elevation or depression might be observed. But before I proceed to the computation and construction of the points in the sections, I shall here abstract the numbers which express the relative elevation of the principal original points in the survey; being the extremes of the lines whose lengths are above abstracted. These few numbers are the results of the calculation of several hundreds of triangles conceived in a vertical position, their bases being either the horizontal lines above-written, or other lines drawn as diagonals between many distant points in the survey, according to the number of vertical angles which had been observed; and of these bases, whether real or imaginary, each generally afforded two vertical triangles, as the angles of elevation and depression were taken alternately at both ends of the lines. It is scarcely necessary to remark, that all these triangles are right-angled, the common base being one of the sides about the right angle, and the other the difference in altitude between the two
Mr. Hutton's Calculations to ascertain given points or extremes of the base; and this difference in altitude is found from the application of this proportion, as radius is to the tangent of the angle of elevation or depression, so is the given base to the altitudinal difference between the two given points, exclusive of the height of the theodolite or other instrument, which was afterwards allowed for. From the resolution of all these triangles, and taking the means of the many corresponding results, were obtained the following numbers, which shew how many feet the points denoted by the letters standing against them are below the level of the point N or the Western cairn. They are all referred to this point N at the Western extremity of the ridge of the hill, because it is the most elevated point in the whole survey.
| o 1184 | γ 2898 | h 2143 | u 1613 | e 2145 |
|--------|--------|--------|--------|--------|
| p 1457 | a 1303 | w 2024 | x 1996 | m 1958 |
| k 480 | b 1313 | l 2006 | s 1964 | m' 322 |
| r 1948 | c 1384 | y 2335 | a 1012 | f' 2246 |
| b'' 1920 | d 1445 | v 2119 | b 823 | t' 2815 |
| α 2898 | f 1904 | t 2114 | c 1364 | k 2835 |
| β 2901 | g 1935 | z 1815 | d 1539 | δ' 172 |
These depressions, and those of several other principal points, were first carefully computed by means of various different bases, as so many places from whence the sections were to commence.
These sections are very numerous, made in all directions from the primitive points before mentioned, and many of them extended to great distances far beyond the bounds of the plan hereunto annexed, so as to include the nearest hills and valleys of the surrounding country. They are mostly made in vertical planes in the manner described in the article of the Phil. Trans. before referred to, excepting some few of them which are level sections in planes parallel to the horizon, and some indeed irregular as being neither vertical nor horizontal. To compute the relative altitude of each point in these sections, it is evident, requires the resolution of two different triangles, viz. a horizontal triangle by which its place in the plan is ascertained, and a vertical triangle of which one side is the elevation or depression of the point. Of these sections there are above 70, containing near 1000 points, whose places in the plan and relative altitudes have been computed: so that the number of triangles, whose numerical resolutions have been performed in the course of this business, amounts to several thousands.
Before the abstract of the computation of the sections, I shall here put down at large the calculation of one of them, to shew the manner in which they have been computed in the readiest and easiest way that occurred to me, preserving at the same time the proper degree of accuracy.
I shall for this purpose select the third section as not containing so many poles as some of the others. This section commences at $s$, and is carried up the hill in a vertical plane, making an angle of $105^\circ$ with the line $rs$. The direction of this plane is here represented by the line $sppp$ making with $rs$ the angle $rsp = 105^\circ$. The points $ppp \&c.$ mark the places of the poles, whose angles of elevation or depression were taken at $s$ with a proper instrument, and they are written in the second column of the table in this example. At $r$ were observed the several horizontal angles, which lines supposed to be drawn from thence made with $rs$, and these are placed in the third column. And since in every triangle $rsp$, the angle $s$ is constant, and the sum of $r$ and $p$ is equal to the constant quantity $75^\circ$: therefore each of the angles $r$, or the numbers in the third column, being subtracted from $75^\circ$, there remains the corresponding angle $p$: and these remainders are placed in the fourth column. Then, since the method of solution is this, as $f.p : f.r :: rs : sp = \frac{f.r}{f.p} \times rs$; and again, as radius $(1) : \text{tang. elev.} :: sp : \text{alt. of } p \text{ above } s = sp \times \text{tang. elev.} = \frac{f.r}{f.p} \times rs \times \text{tang. elev.}$ Or in logarithms
$$\begin{array}{c}
\text{From} \\
\text{Take} \\
\text{Leaves}
\end{array}$$
$$\begin{array}{c}
- & 180^\circ \\
\angle s = & 105^\circ \\
\angle r + \angle p = & 75^\circ
\end{array}$$
the mean Density of the Earth.
\[ f_r - f_p + r_s + \text{tang. elev.} = \log. \text{ of the altitude of the point.} \]
Wherefore having taken, from a table, the fines of \( r \) and \( p \), and placed them in the fifth and sixth columns, subtract the latter from the former, and write the remainders in the next or seventh column; to these add the constant logarithm of \( r_s \), and write the sums in the eighth column; take out then the tangents of the angles in the second column, and having placed them in the ninth column, add together the adjacent numbers of the eighth and ninth columns, placing the sums in the tenth column, which being the logarithms of the altitudes or depressions of the points \( p \), take the corresponding numbers from a table of logarithms, and write them in the eleventh or last column, for those altitudes or depressions with respect to the point \( s \), with the height of the theodolite included, and which is afterwards allowed for, its height being generally about \( 4\frac{1}{2} \) or \( 4\frac{3}{4} \) feet. In the second column \( D \) denotes depression and \( E \) elevation; in the last column \( D \) denotes depression and \( A \) altitude.
| No. of Poles | Angles of Dep. and Elev. at s. | Horiz. Angs. alt. R. | Sines of R. | Sines of p. | Sin. R. fin. p. | Sin. R. fin. p. + rs. or s. θ. | Tang. of Dep. and Elev. | Sum of 8 and 9, or Lag. alt. above | Sum of 8 and 9, or Lag. alt. below |
|-------------|-------------------------------|---------------------|------------|------------|---------------|---------------------------------|--------------------------|----------------------------------|----------------------------------|
| 1 | ° 3 27D | ° 19 11 | 55 49 | 9°51666 | 9°91763 | 9°59903 | 2°74843 | 8°78022 | 1°5°865 |
| 2 | 2 36E | 30° 24 | 44 36 | 9°70418 | 9°84643 | 9°85775 | 3°00715 | 8°65715 | 1°5°865 |
| 3 | 4 33 | 38° 28 | 36 32 | 9°79388 | 9°77473 | 9°81910 | 3°16852 | 8°9080 | 2°66930 |
| 4 | 6 12 | 44° 10 | 30 50 | 9°84308 | 9°70973 | 9°13335 | 3°28275 | 9°3597 | 2°31872 |
| 5 | 7 51 | 47° 55 | 27 5 | 9°87050 | 9°61828 | 9°25222 | 3°40162 | 9°13948 | 2°54110 |
| 6 | 10 38 | 51° 22 | 23 38 | 9°89574 | 9°60302 | 9°2972 | 3°44212 | 9°27357 | 2°71569 |
| 7 | 12 20 | 53° 9 | 21 51 | 9°90320 | 9°57075 | 9°3345 | 3°48185 | 9°33974 | 2°82159 |
| 8 | 13 46 | 54° 43 | 20 17 | 9°91185 | 9°53991 | 9°37194 | 3°52134 | 9°38918 | 2°91052 |
| 9 | 15 43 | 56° 21 | 18 39 | 9°92035 | 9°59486 | 9°41549 | 3°56489 | 9°44933 | 3°01422 |
| 10 | 17 35 | 57° 47 | 17 13 | 9°92739 | 9°47127 | 9°45612 | 3°60552 | 9°50092 | 3°10644 |
| 11 | 18° | 58° 58 | 16 2 | 9°93291 | 9°44122 | 9°49169 | 3°64109 | 9°51178 | 3°15287 |
Thus
Thus then every line in the table contains the solutions of the two triangles, the one horizontal and the other vertical; used in finding the altitude of each point or pole in the section. The addition of the constant logarithm of the base \(rs\) to the logarithms in the seventh column, is most easily performed by writing it on the bottom of a little slip of paper, and so sliding it down successively over each of those numbers, and in that position adding them together, and placing the sums immediately opposite in the next column.
And in this manner were computed the relative altitudes of the points in the other vertical sections; excepting two or three cases, in which the constant angle formed by the section and the base was a right angle; and one case in which the vertical angles were not taken at the beginning of the section line, but at the other end of the base line where the horizontal angles were also observed. It may be necessary, therefore, to insert and explain an example of each of these cases, and the more so as they point out the properest means of measuring these sections so as to save most part of the labour in the computation, in which the trouble chiefly consists.
Of the case of the right angle, the first section is an instance, where also \(rs\) is the base as before, and the angle \(rsp\) being = 90°.
In this form there are three columns less than in the former, by which it happens, that about one-third of the labour is saved. The method of solution is thus; as \( r \) (radius) : tang. \( R \) :: \( RS : SP = RS \times t.R \); and again, as \( r : \text{tang.} s \) (vertical angle) :: \( SP : SP \times t.s = RS \times t.R \times t.s \). Or, in logarithms, \( \log. RS + t.R + t.s = \log. \) of the vertical perpendicular: and by this theorem, it is evident, the columns of this table are constructed.
But
But nearly the same saving in the great labour of computation would be made if the vertical and horizontal angles had both been taken at the end of the base farthest from the beginning of the section. And this method would also be much the easiest in making the survey on the ground, as there would then need only one observer with an instrument to measure both horizontal and vertical angles; and any person, without an instrument, could direct in a line the person who moves and places the poles, or he may even direct himself after his first pole has been placed, by means of a back object, as is commonly done in land surveying.
Of this kind there happens to have been one section taken, proceeding from G, and making with GP an angle of $85^\circ$, P being the Northern observatory, and where both the bearings and depressions of the points p in the section line were observed.
\[
\begin{align*}
\text{Log.} & \\
PG & \quad 3.68262 \\
f.G & \quad 9.99834 \\
\text{Sum} & \quad 3.68096
\end{align*}
\]
which is a constant number from which the fines of p in the fifth column are to be deducted.
Here it is evident is a saving of two of the most laborious columns in the table. This happens because that in every triangle \( pGp \) there are now constant those two parts which are used in the proportion made use of in the calculation, viz. \( pG \) and the angle \( G \). For then it is, as \( f.p : f.G :: pG : p.p \), or \( \log_p = \log_p pG + f.G - f.p \); so that the sum of the logarithms of \( pG \) and sine of \( \angle G \) is a constant number, from which the numbers in the fifth column are to be subtracted, to find those in the sixth column. The rest of the work is the same as in the first example.
As to the irregular sections, the computation of them differs so little in manner from that of the usual vertical sections,
sections, that an example of it is unnecessary: and the few horizontal sections need no computation, but only an allowance for the height of the theodolite.
In the following abstract of the results of the computation of the sections, the first column contains the number of the pole, the second and third the vertical and horizontal angles, and the last the difference of altitude in feet, between the foot of each pole and the point from whence the vertical angles were observed, after making the allowance for the height of the theodolite above the ground. At the end of this abstract is a plate of the figures referring to the number of the section, shewing the direction in which it was carried, with the degrees and minutes in the angle formed by it and the base line.
Mr. Hutton's Calculations to ascertain
| Pole | Vert. ∠'s at s. | Bearings at R. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 5° 16½ D | 10° 0 | 18 D |
| 2 | 0° 30 E | 31° 35 | 12 A |
| 3 | 4° 15 | 41° 56 | 99 |
| 4 | 6° 14¼ | 49° 25 | 185 |
| 5 | 8° 16 | 55° 51½ | 307 |
| 6 | 10° 13 | 59° 57½ | 444 |
| 7 | 11° 37 | 62° 56½ | 572 |
| 8 | 12° 25 | 65° 3¾ | 673 |
| 9 | 13° 21 | 66° 41½ | 782 |
| 10 | 14° 10 | 67° 36¼ | 869 |
| 11 | 15° 17 | 68° 42½ | 994 |
| 12 | 17° 46 | 70° 58 | 1315 |
| 13 | 19° 33 | 72° 48 | 1623 |
| 14 | 20° 6 | 74° 30 | 1866 |
| Pole | Vert. ∠'s at s. | Bearings at R. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 3° 40 D | 20° 46 | 30 D |
| 2 | 1° 26 E | 32° 42 | 28 A |
| 3 | 4° 20 | 42° 8 | 103 |
| 4 | 6° 21 | 49° 30 | 192 |
| 5 | 9° 58½ | 59° 2 | 429 |
| 6 | 11° 50½ | 62° 30 | 591 |
| 7 | 12° 52 | 65° 2 | 721 |
| 8 | 13° 22 | 65° 51 | 780 |
| 9 | 13° 36½ | 66° 9 | 806 |
| Pole | Vert. ∠'s at s. | Bearings at R. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 7° 42 D | 13° 8 | 39 D |
| 2 | 1° 2 E | 31° 35 | 19 A |
| 3 | 4° 4 | 40° 24 | 80 |
| 4 | 5° 36 | 48° 36 | 137 |
| 5 | 6° 55 | 56° 56 | 215 |
| Pole | Vert. ∠'s at x. | Bearings at s. | Diff. of Alt. |
|------|----------------|----------------|---------------|
| 1 | 3 46 D | 15 6 | 38 D |
| 2 | 0 14 E | 24 23 | 9 A |
| 3 | 2 16 | 36 14 | 60 |
| 4 | 3 29 | 46 5 | 112 |
| 5 | 4 21 | 54 24 | 163 |
| 6 | 5 48 | 63 55 | 258 |
| 7 | 7 2 | 71 30 | 361 |
| 8 | 9 3 | 76 30 | 513 |
| 9 | 11 25 | 81 18 | 720 |
| 10 | 12 55 | 84 16 | 874 |
| 11 | 13 54 | 87 22 | 1017 |
| 12 | 14 51 | 90 30 | 1182 |
| 13 | 15 9 | 93 0 | 1295 |
| Pole | Vert. ∠'s at x. | Bearings at s. | Diff. of Alt. |
|------|----------------|----------------|---------------|
| 1 | 7 30 D | 9 30 | 60 D |
| 2 | 1 49 | 15 52 | 24 |
| 3 | 2 5 E | 20 49 | 52 A |
| 4 | 4 30 | 24 48 | 135 |
| 5 | 5 41 | 28 23 | 208 |
| 6 | 7 13 | 30 27 | 297 |
| 7 | 8 14 | 32 29 | 380 |
| 8 | 9 24 | 33 41 | 465 |
| 9 | 10 26 | 35 0 | 558 |
Mr. Hutton's Calculations to ascertain
| Pole | Vert. ∠'s at A. | Bearings at B. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 17° 56' 1/2 D | 37° 12' | 516 D |
| 2 | 10° 12' 1/2 | 41° 50' | 359 |
| 3 | 5° 41' | 44° 48' | 229 |
| 4 | 2° 25' | 47° 2' | 107 |
| 5 | 0° 29' | 49° 6' | 20 |
| 6 | 1° 12' E | 51° 0' | 74 A |
| 7 | 3° | 52° 35' 1/2 | 198 |
| 8 | 5° | 54° 44' 1/2 | 378 |
| 9 | 6° 18' | 55° 56' 1/2 | 520 |
| 10 | 7° 2' | 56° 47' 1/2 | 620 |
| 11 | 8° 52' | 57° 24' | 821 |
| 12 | 10° 10' | 57° 55' 1/2 | 986 |
| 13 | 11° 24' | 58° 28' | 1160 |
| 14 | 12° 20' 1/2 | 58° 58' | 1316 |
N.B. The place of this last pole would seem to be the same as N the Western cairn, as the section was directed through it. But then the last number 1316 is too great; as, from all the other measures, the diff. in alt.
between A and N is only 1303 feet. This diff. of 13 feet seems to be caused by the last bearing being about 7' too great, for in other places this angle is only 58° 51'. And indeed many other angles taken at the same time with the above seem to be much wrong, as they greatly differ from corresponding ones taken at other times.
Such differences among corresponding angles I often met with in the measures contained in the books of the survey, and it required much care to detect them, and trouble to reconcile them.
SECTION 9.
| Pole | Bearings at A. | Bearings at B. |
|------|----------------|---------------|
| 1 | 132° 21' 1/2 | 36° 45' 1/2 |
| 2 | 130° 27' | 37° 34' 1/2 |
| 3 | 127° 26' | 39° 8' |
| 4 | 124° 25' 1/2 | 40° 18' |
| 5 | 119° 6' 1/2 | 43° 19' |
| 6 | 111° 37' | 48° 7' |
| 7 | 103° 45' 1/2 | 52° 58' |
| 8 | 95° 25' | 58° 50' |
| 9 | 85° 55' | 67° 54' |
| 10 | 78° 2' 1/2 | 75° 58' 1/2 |
| 11 | 73° 14' | 82° 30' |
| 12 | 71° 41' 1/2 | 86° 21' |
| 13 | 70° 38' 1/2 | 89° 33' |
| 14 | 69° 39' | 92° 23' |
This is a horizontal or level section through A, and therefore each point is 42 feet (the height of the theodolite) above that point.
### SECTION I0.
| Pole | Vert. ∠'s at A. | Bearings at B. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 19 41 D | 51 12 | 531 D |
| 2 | 13 52 | 57 22½ | 426 |
| 3 | 8 7½ | 62 56 | 295 |
| 4 | 2 30 | 68 47 | 106 |
| 5 | 0 59 | 73 55½ | 47 |
| 6 | 0 22½ E | 76 52 | 27 A |
| 7 | 0 59 | 79 14½ | 70 |
| 8 | 1 39 | 81 7 | 124 |
| 9 | 2 25 | 82 42 | 194 |
| 10 | 3 5 | 84 12 | 266 |
| 11 | 3 36 | 85 41½ | 338 |
| 12 | 3 48 | 86 27 | 373 |
### SECTION I2.
| Pole | Bearings at D. | Bearings at C. |
|------|----------------|----------------|
| 1 | 68 14½ | 102 6 |
| 2 | 71 16 | 98 23 |
| 3 | 73 55 | 94 56 |
| 4 | 79 0 | 89 33 |
| 5 | 85 28 | 82 49 |
| 6 | 91 19 | 76 48 |
| 7 | 97 5 | 71 8 |
| 8 | 102 21 | 66 20 |
| 9 | 108 18 | 60 36 |
| 10 | 113 33 | 56 15 |
| 11 | 117 12 | 53 17 |
| 12 | 119 16 | 52 4 |
| 13 | 119 53 | 52 1 |
Each of these poles is five feet above D, the feet being horizontal and taken at that point.
### SECTION II.
| Pole | Vert. ∠'s at D. | Bearings at C. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 12 56 D | 43 18 | 231 D |
| 2 | 9 50 | 62 4 | 334 |
| 3 | 5 15 | 67 9 | 242 |
| 4 | 2 25 | 69 51½ | 135 |
| 5 | 0 59 E | 71 20 | 70 A |
| 6 | 3 0 | 72 13 | 220 |
| 7 | 4 31 | 73 9 | 363 |
| 8 | 6 7 | 73 53 | 533 |
| 9 | 6 59 | 74 27 | 652 |
| 10 | 7 53 | 74 52 | 777 |
| 11 | 8 49 | 75 9½ | 904 |
| 12 | 9 27 | 75 42 | 1048 |
### SECTION I3.
| Pole | Vert. ∠'s at G. | Bearings at F. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 3 7 D | 75 2 | 92 D |
| 2 | 0 12 | 79 46 | 2 |
| 3 | 4 1 E | 91 55 | 221 A |
| 4 | 6 8 | 95 0 | 385 |
| 5 | 7 49 | 96 34 | 530 |
| 6 | 9 50 | 99 27 | 789 |
| 7 | 10 52 | 100 10 | 915 |
Vol. LXVIII.
Mr. Hutton's Calculations to ascertain
| Pole | Vert. ∠'s at G. | Bearings at F. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 2° 3' E | 97° 5 | 73 A |
| 2 | 2° 35 | 99° 14 | 95 |
| 3 | 3° 52 | 109° 39 | 182 |
| 4 | 5° 58 | 117° 53 | 384 |
| 5 | 6° 25 | 119° 8 | 440 |
| 6 | 8° 31 | 120° 34 | 633 |
| 7 | 10° 14 | 121° 51 | 827 |
| 8 | 11° 29 | 122° 41 | 985 |
| 9 | 12° 30 | 123° 34 | 1149 |
| 10 | 13° 4 | 124° 15 | 1272 |
| 11 | 13° 14 | 124° 41 | 1339 |
SECTION 15.
| Pole | Vert. ∠'s at G. | Bearings at H. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 9° 46 D | 14° 14 | 172 D |
| 2 | 3° 32 | 23° 9 | 102 |
| 3 | 0° 45 | 31° 7 | 27 |
| 4 | 1° 43 E | 36° 15 | 94 A |
SECTION 16.
| Pole | Vert. ∠'s at H. | Bearings at G. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 7° 22 D | 21° 37 | 189 D |
| 2 | 6° 11 | 34° 32 | 248 |
| 3 | 4° 1 | 43° 23 | 201 |
| 4 | 0° 9 | 48° 20 | 4 |
| 5 | 1° 43 E | 54° 38 | 119 A |
| 6 | 3° 35 | 60° 8 | 275 |
| 7 | 5° 14 | 65° 8 | 446 |
| 8 | 6° 49 | 70° 13 | 653 |
| 9 | 8° 1 | 73° 19 | 828 |
| 10 | 8° 41 | 76° 39 | 977 |
| 11 | 9° 18 | 78° 38 | 1104 |
| 12 | 10° 0 | 80° 22 | 1246 |
| 13 | 10° 44 | 82° 0 | 1403 |
| 14 | 11° 31 | 83° 24 | 1572 |
| 15 | 13° 8 | 84° 43 | 1874 |
There seems to be some general error in this section, as the depressions and altitudes are utterly incompatible with those of all the other neighbouring points in the plan.
### SECTION 17
| Pole | Vert. ∠'s at H. | Bearings at G. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 12 15 D | 16 13 | 242 D |
| 2 | 7 11 | 25 6 | 216 |
| 3 | 4 19 | 33 3 | 171 |
| 4 | 1 49 | 38 20 | 82 |
| 5 | 1 59 E | 45 24 | 119 A |
| 6 | 3 38 | 50 3 | 243 |
| 7 | 5 10 | 53 33 | 377 |
| 8 | 7 10 | 57 3 | 572 |
| 9 | 8 17 | 60 46 | 731 |
| 10 | 8 45 | 62 58 | 820 |
| 11 | 9 53 | 66 56 | 1038 |
| 12 | 10 24 | 69 0 | 1161 |
| 13 | 10 45 | 70 35 | 1260 |
| 14 | 11 31 | 72 15 | 1424 |
| 15 | 12 10 | 73 55 | 1590 |
| 16 | 12 31 | 75 7 | 1703 |
### SECTION 18
| Pole | Vert. ∠'s at H. | Bearings at W. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 13 35 D | 21 40 | 186 D |
| 2 | 9 17½ | 43 28 | 263 |
| 3 | 5 21 | 55 3 | 203 |
| 4 | 2 4 | 63 8 | 95 |
| 5 | 1 23 E | 69 50 | 84 A |
| 6 | 3 37 | 73 24 | 238 |
| 7 | 5 25 | 76 0 | 386 |
### SECTION 19
| Pole | Vert. ∠'s at W. | Bearings at L. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 18 8 D | 14 51 | 168 D |
| 2 | 16 28 | 31 38 | 388 |
| 3 | 11 53 | 39 42 | 394 |
| 4 | 7 47 | 42 48 | 292 |
| 5 | 4 11 | 46 12 | 180 |
| 6 | 2 10 | 48 6 | 100 |
| 7 | 0 31 | 50 16 | 23 |
| 8 | 2 24 E | 52 45 | 148 A |
| 9 | 5 55 | 54 48 | 402 |
| 10 | 8 48 | 56 30 | 657 |
### SECTION 20
| Pole | Vert. ∠'s at W. | Bearings at L. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 22 45 D | 7 37 | 98 D |
| 2 | 19 0 | 16 20 | 179 |
| 3 | 18 41 | 31 8 | 367 |
| 4 | 14 20 | 41 54 | 412 |
| 5 | 9 41 | 48 4 | 341 |
Mr. Hutton's Calculations to ascertain
| Pole | Vert. ∠'s at L. | Bearings at W. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 23° 59' D | 21° 18' | 295° D |
| 2 | 19° 25' | 33° 12' | 377 |
| 3 | 14° 37' | 48° 26' | 448 |
| 4 | 10° 5' | 56° 7' | 383 |
| 5 | 5° 9 | 62° 51' | 237 |
| 6 | 1° 2 | 69° 24' | 56 |
| 7 | 2° 2 E | 71° 36' | 133° A |
| 8 | 4° 22 | 73° 2 | 297 |
| 9 | 6° 6 | 75° 22' | 455 |
**SECTION 21**
| Pole | Vert. ∠'s at a. | Bearings at b. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 1° 15' 2 D | 4° 33' | 34° D |
| 2 | 0° 51 | 15° 15' | 4 |
| 3 | 0° 10 | 35° 37' | 0 |
| 4 | 3° 7 E | 41° 53' | 97° A |
| 5 | 6° 43 | 42° 34' | 209° |
| 6 | 7° 36 | 43° 30' | 244 |
| 7 | 9° 50 | 45° 48' | 342 |
| 8 | 11° 35 | 59° 5 | 664 |
**SECTION 22**
| Pole | Vert. ∠'s at a. | Bearings at b. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 7° 16' D | 5° 2 | 20° D |
| 2 | 2° 49 E | 10° 20' | 25° A |
| 3 | 3° 33 | 15° 13' | 44 |
| 4 | 4° 18 | 29° 34' | 120 |
| 5 | 7° 0 | 32° 15' | 222 |
| 6 | 9° 1 | 35° 3 | 331 |
| 7 | 10° 26 | 38° 52' | 471 |
| 8 | 12° 19 | 41° 46' | 657 |
**SECTION 23**
| Pole | Vert. ∠'s at a. | Bearings at b. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 1° 14' 40' | 43° 34' | 874 |
| 2 | 1° 14' 45' top | 43° 35' | 874 |
**SECTION 24**
| Pole | Vert. ∠'s at b. | Bearings at a. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 3° 27' D | 10° 29' | 29° D |
| 2 | 0° 32 E | 21° 9 | 12° A |
| 3 | 2° 58 | 32° 57' | 64 |
| 4 | 4° 40 | 58° 53' | 133 |
| 5 | 5° 14 | 76° 22' | 173 |
| 6 | 6° 24 | 121° 5 | 34° |
| 7 | 7° 50 | 125° 12 | 448 |
| 8 | 8° 27 | 130° 23 | 546 |
| 9 | 8° 55 | 134° 57 | 664 |
| 10 | 9° 6 | 136° 15 | 712 |
### SECTION 25.
| Pole | Vert. ∠'s at b | Bearings at a | Diff. of Alt. |
|------|----------------|--------------|---------------|
| 1 | 15° 39' D | 8° 14' | 74° D |
| 2 | 1° 47' E | 52° 24' | 54° A |
| 3 | 3° 9' | 63° 13' | 115° |
| 4 | 5° 51' | 70° 37' | 248° |
| 5 | 9° 11' | 81° 35' | 505° |
| 6 | 11° 21' | 85° 10' | 691° |
| 7 | 12° 44' | 87° 45' | 802° |
### SECTION 26.
| Pole | Vert. ∠'s at b | Bearings at a | Diff. of Alt. |
|------|----------------|--------------|---------------|
| 1 | 17° 53' D | 6° 35' | 64° D |
| 2 | 0° 26' | 48° 5' | 9° |
| 3 | 2° 1' A | 60° 8' | 93° A |
| 4 | 3° 4' | 62° 45' | 151° |
| 5 | 4° 40' | 66° 15' | 256° |
| 6 | 6° 20' | 68° 29' | 374° |
| 7 | 8° 10' | 71° 57' | 55° |
### SECTION 27.
| Pole | Vert. ∠'s at b | Bearings at a | Diff. of Alt. |
|------|----------------|--------------|---------------|
| 1 | 4° 30' E | 59° 25' | 280° A |
| 2 | 3° 33' | 57° 12' | 204° |
| 3 | 1° 47' | 54° 11' | 92° |
| 4 | 0° 10' D | 50° 15' | 3° D |
| 5 | 1° 22' | 46° 54' | 47° |
### SECTION 28.
| Pole | Vert. ∠'s at T | Bearings at v | Diff. of Alt. |
|------|----------------|--------------|---------------|
| 1 | 1° 23' D | 16° 36' | 142° D |
| 2 | 9° 26' | 23° 56' | 195° |
| 3 | 5° 23' | 28° 45' | 136° |
| 4 | 4° 14' | 32° 24' | 120° |
| 5 | 2° 42' | 37° 46' | 96° |
| 6 | 1° 34' | 41° 20' | 62° |
| 7 | 0° 35½ | 45° 24' | 24° |
| 8 | 1° 30' E | 48° 52' | 89° A |
| 9 | 3° 30' | 51° 4' | 220° |
| 10 | 4° 20' | 54° 22' | 307° |
| 11 | 4° 49' | 56° 11' | 367° |
| 12 | 5° 15' | 58° 41' | 443° |
| 13 | 6° 0' | 60° 4' | 537° |
| 14 | 6° 47' | 62° 4' | 664° |
| 15 | 7° 24' | 63° 33' | 778° |
| 16 | 8° 16' | 64° 46' | 924° |
| 17 | 8° 46' | 65° 44' | 1030° |
### SECTION 29.
| Pole | Vert. ∠'s at T | Bearings at v | Diff. of Alt. |
|------|----------------|--------------|---------------|
| 1 | 8° 49' D | 26° 9' | 182° D |
| 2 | 6° 51' | 30° 7' | 163° |
Mr. Hutton's Calculations to ascertain
| Pole | Vert. ∠'s at T. | Bearings at v. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 3° 26' E | 8° 1' 6" | 353 A |
| 2 | 3° 10' | 7° 9' 53" | 313 |
| 3 | 2° 55' | 7° 7' 7" | 263 |
| 4 | 2° 46' | 7° 5' 2" | 233 |
| 5 | 2° 35' | 7° 2' 45" | 204 |
| 6 | 2° 26' | 7° 0' 16" | 179 |
| 7 | 1° 20' | 6° 6' 28" | 91 |
| 8 | 0° 19' | 6° 2' 32" | 23 |
| 9 | 0° 49' D | 5° 7' 58" | 37 D |
| 10 | 2° 33' | 5° 1' 54" | 107 |
| 11 | 4° 23' | 4° 3' 37" | 151 |
| 12 | 6° 21' | 3° 4' 48" | 176 |
| Pole | Vert. ∠'s at Y. | Bearings at v. | Diff. of Alt. |
|------|-----------------|---------------|--------------|
| 1 | 1° 11' 18" D | 1° 10' 40" | 132 D |
| 2 | 5° 29' | 1° 15' 43" | 93 |
| 3 | 2° 56' | 1° 19' 13" | 60 |
| 4 | 1° 42' | 2° 22' 49" | 40 |
| 5 | 0° 42' | 2° 28' 23" | 20 |
| 6 | 0° 16' | 3° 31' 43" | 5 |
| 7 | 0° 30' E | 3° 36' 10" | 28 A |
| 8 | 1° 33' | 4° 40' 41" | 90 |
| 9 | 2° 13' | 4° 45' 0" | 146 |
| 10 | 2° 41' | 4° 47' 18" | 190 |
| 11 | 2° 54' | 5° 50' 47" | 231 |
| 12 | 3° 13' | 5° 53' 12" | 278 |
| 13 | 3° 39' | 5° 56' 0" | 349 |
| 14 | 4° 53' | 5° 58' 51" | 519 |
| 15 | 5° 42' | 6° 60' 38" | 650 |
| 16 | 6° 0' | 6° 62' 8" | 728 |
| 17 | 6° 27' | 6° 63' 23" | 825 |
| 18 | 6° 41' | 6° 64' 21" | 892 |
| 19 | 6° 55' | 6° 65' 50" | 988 |
### SECTION 32
| Pole | Vert. ∠'s at Y. | Bearings at V. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 7° 9' E | 51° 46' | 812 A |
| 2 | 6° 55' | 50° 50' | 751 |
| 3 | 6° 7' | 49° 45' | 631 |
| 4 | 5° 0' | 46° 51' | 452 |
| 5 | 4° 25' | 45° 30' | 377 |
| 6 | 3° 29' | 43° 37' | 275 |
| 7 | 3° 5' | 40° 32' | 214 |
| 8 | 2° 17' | 38° 26' | 146 |
| 9 | 1° 21' | 35° 57' | 80 |
| 10 | 0° 52' | 33° 0' | 47 |
| 11 | 0° 3' | 29° 47' | 7 |
| 12 | 1° 3° D | 26° 24' | 33° D |
| 13 | 2° 3' | 23° 40' | 60 |
| 14 | 3° 21' | 18° 30' | 73 |
| 15 | 7° 54' | 13° 4' | 121 |
### SECTION 34
| Pole | Vert. ∠'s at T. | Bearings at Z. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 4° 43' E | 84° 3' | 471 A |
| 2 | 4° 8' | 79° 45' | 386 |
| 3 | 3° 31' | 76° 0' | 311 |
| 4 | 2° 45' | 71° 30' | 229 |
| 5 | 2° 0' | 66° 22' | 156 |
| 6 | 1° 8' | 60° 49' | 84 |
| 7 | 0° 19' | 53° 20' | 25 |
| 8 | 0° 55° D | 46° 47' | 46° D |
| 9 | 1° 29' | 40° 15' | 69 |
| 10 | 1° 59' | 34° 52' | 83 |
| 11 | 3° 41' | 26° 15' | 127 |
| 12 | 5° 27' | 21° 55' | 165 |
| 13 | 6° 0' | 17° 16' | 150 |
| 14 | 9° 5' | 9° 56' | 143 |
| 15 | 14° 6' | 4° 32' | 109 |
### SECTION 33
| Pole | Vert. ∠'s at T. | Bearings at Z. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 13° 6° D | 4° 6' | 108 D |
| 2 | 9° 15' | 8° 4' | 139 |
| 3 | 7° 32' | 14° 53' | 187 |
| 4 | 5° 10' | 21° 28' | 167 |
| 5 | 3° 55' | 26° 54' | 148 |
| 6 | 2° 36' | 36° 19' | 119 |
| 7 | 1° 30' | 45° 32' | 78 |
| 8 | 0° 40' | 56° 5' | 37 |
### SECTION 35
| Pole | Vert. ∠'s at T. | Bearings at Z. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 15° 16° D | 1° 58' | 86 D |
| 2 | 9° 36' | 5° 3' | 126 |
Mr. Hutton's Calculations to ascertain
| Pole | Vert. ∠'s at u. | Bearings at z. | Diff. of Alt. |
|------|----------------|----------------|---------------|
| 1 | 3 53 E | 63 50 | 345 A |
| 2 | 3 27 | 60 54 | 286 |
| 3 | 1 55 | 57 45 | 150 |
| 4 | 0 16 | 54 3 | 23 |
| 5 | 1 14 D | 50 40 | 74 D |
| 6 | 2 53 | 47 28 | 166 |
| 7 | 4 45 | 43 0 | 247 |
| 8 | 6 25 | 38 49 | 300 |
| 9 | 8 40 | 33 14 | 348 |
| 10 | 12 6 | 28 25 | 420 |
| 11 | 16 36 | 23 43 | 419 |
SECTION 36.
| Pole | Vert. ∠'s at u. | Bearings at z. | Diff. of Alt. |
|------|----------------|----------------|---------------|
| 1 | 17 42 D | 2 45 | 60 D |
| 2 | 15 55 | 22 26 | 465 |
| 3 | 12 39 | 25 13 | 404 |
| 4 | 8 51 | 29 37 | 333 |
| 5 | 7 23 | 33 36 | 318 |
| 6 | 3 51 | 40 33 | 205 |
| 7 | 1 22 | 44 45 | 80 |
| 8 | 1 3 E | 48 26 | 77 A |
| 9 | 2 54 | 50 52 | 218 |
| 10 | 4 22 | 52 45 | 347 |
| 11 | 5 37 | 54 40 | 471 |
| 12 | 6 30 | 55 30 | 559 |
SECTION 37.
| Pole | Vert. ∠'s at u. | Bearings at z. | Diff. of Alt. |
|------|----------------|----------------|---------------|
| 1 | 3 53 E | 63 50 | 345 A |
| 2 | 3 27 | 60 54 | 286 |
| 3 | 1 55 | 57 45 | 150 |
| 4 | 0 16 | 54 3 | 23 |
| 5 | 1 14 D | 50 40 | 74 D |
| 6 | 2 53 | 47 28 | 166 |
| 7 | 4 45 | 43 0 | 247 |
| 8 | 6 25 | 38 49 | 300 |
| 9 | 8 40 | 33 14 | 348 |
| 10 | 12 6 | 28 25 | 420 |
| 11 | 16 36 | 23 43 | 419 |
SECTION 38.
| Pole | Vert. ∠'s at u. | Bearings at z. | Diff. of Alt. |
|------|----------------|----------------|---------------|
| 1 | 15 9 D | 20 37 | 405 D |
| 2 | 11 30 | 31 39 | 438 |
| 3 | 9 23 | 36 7 | 398 |
| 4 | 6 46 | 40 30 | 314 |
| 5 | 5 27 | 46 10 | 283 |
| 6 | 3 32 | 52 5 | 204 |
| 7 | 2 49 | 55 9 | 171 |
| 8 | 1 49 | 61 25 | 122 |
### SECTION 39.
| Pole | Vert. ∠'s at U. | Bearings at Z. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 4° 28' D | 69° 7' | 312° D |
| 2 | 5° 35' | 59° 8' | 345° |
| 3 | 6° 30' | 52° 3' | 366° |
| 4 | 8° 9' | 45° 25' | 418° |
| 5 | 10° 32' | 38° 30' | 485° |
| 6 | 12° 47' | 34° 3' | 544° |
| 7 | 13° 47' | 29° 27' | 531° |
### SECTION 40.
| Pole | Vert. ∠'s at P. | Bearings at P. | Diff. of Alt. |
|------|-----------------|----------------|--------------|
| 1 | 8° 45' D | 9° 15' | 735° D |
| 2 | 8° 46' | 16° 25' | 750° |
| 3 | 9° 58' | 27° 16' | 906° |
| 4 | 8° 38' | 30° 52' | 804° |
| 5 | 7° 6' | 34° 30' | 681° |
| 6 | 5° 23' | 37° 55' | 533° |
### SECTION 41.
| Pole | Vert. ∠'s at O. | Bearings at A. | Diff. of Alt. |
|------|-----------------|----------------|--------------|
| 1 | 10° 41' D | 23° 33' | 510° D |
| 2 | 11° 31' | 32° 36' | 769° |
| 3 | 8° 52' | 37° 23' | 682° |
| 4 | 5° 38' | 44° 24' | 524° |
| 5 | 4° 52' | 46° 39' | 480° |
| 6 | 4° 6' | 51° 8' | 455° |
### SECTION 42.
| Pole | Vert. ∠'s at M. | Bearings at S. | Diff. of Alt. |
|------|-----------------|----------------|--------------|
| 1 | 0° 17' D | 73° 48' | 0° |
| 2 | 3° 25' E | 80° 57' | 72° A |
| 3 | 4° 52' | 85° 4' | 126° |
| 4 | 6° 52' | 88° 53' | 232° |
| 5 | 8° 33' | 90° 25' | 332° |
| 6 | 10° 2' | 91° 34' | 439° |
| 7 | 11° 44' | 93° 0' | 610° |
| 8 | 13° 7' | 94° 0' | 787° |
| 9 | 15° 10' | 94° 34' | 1002° |
| 10 | 17° 21' | 95° 13' | 1292° |
| 11 | 18° 38' | 95° 37' | 1506° |
| 12 | 19° 36' | 96° 2' | 1736° |
| 13 | 19° 38' | 96° 0' | 1727° |
### SECTION 43.
| Pole | Vert. ∠'s at O. | Bearings at A. | Diff. of Alt. |
|------|-----------------|----------------|--------------|
| 1 | 12° 3' D | 31° 52' | 759° D |
| 2 | 10° 54' | 36° 42' | 776° |
**Vol. LXVIII.**
**4 Y**
Mr. Hutton's Calculations to ascertain
| Pole | Vert. ∠'s at R. | Bearings at S. | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 3° 41' D | 6° 37' | 32° D |
| 2 | 1° 18' | 10° 10' | 21 |
| 3 | 0° 51' E | 11° 21' | 25° A |
| 4 | 0° 52' | 12° 20' | 30 |
| 5 | 1° 4' | 16° 25' | 70 |
| 6 | 2° 3' | 16° 52' | 143 |
| 7 | 3° 6' | 17° 27' | 243 |
| 8 | 4° 38' | 18° 0' | 413 |
| 9 | 5° 14' | 18° 6' | 478 |
| 10 | 5° 48' | 18° 11' | 530 |
| 11 | 6° 28' | 18° 27' | 647 |
| Pole | Vert. ∠'s at D. | Bearings at C. | Diff. of Alt. |
|------|-----------------|----------------|--------------|
| 1 | 8° 50' D | 9° 20' | 201° D |
| 2 | 7° 24' | 11° 4' | 260 |
| 3 | 6° 36' | 11° 46' | 272 |
| 4 | 6° 9' | 11° 22' | 275 |
| 5 | 5° 24' | 11° 44' | 298 |
| 6 | 5° 4' | 12° 17' | 315 |
| 7 | 4° 16' | 12° 24' | 292 |
| 8 | 3° 27' | 12° 13' | 255 |
| 9 | 2° 23' | 12° 17' | 197 |
| 10 | 1° 17' | 12° 55' | 113 |
| 11 | 0° 44' E | 12° 50' | 82° A |
| 12 | 1° 17' | 12° 8' | 146 |
| Pole | Vert. ∠'s at A. | Bearings at R. | Diff. of Alt. |
|------|-----------------|----------------|--------------|
| 1 | 2° 15' D | 3° 14' | 122° D |
| 2 | 2° 0' | 3° 57' | 116 |
| 3 | 1° 22' | 3° 35' | 100 |
| Pole | Vert. ∠'s at T. | Bearings at V. | Diff. of Alt. |
|------|-----------------|----------------|--------------|
| 1 | 1° 42' D | 1° 45' | 152° D |
| 2 | 6° 6' | 2° 48' | 145 |
| 3 | 3° 19' | 3° 30' | 111 |
| Pole | Vert. ∠'s at v. | Bearings at v. | Diff. of Alt. |
|------|----------------|----------------|---------------|
| 1 | 9° 6° D | 17° 3° | 164° D |
| 2 | 7° | 30° 56° | 236 |
| 3 | 3° 22° | 44° 36° | 170 |
| 4 | 1° 16° | 47° 23° | 66 |
| 5 | 0° 5° | 49° 52° | 0 |
| 6 | 2° 46° E | 58° 6° | 214 A |
| 7 | 4° 57° | 64° 34° | 457 |
| 8 | 5° 53° | 67° 2° | 585 |
| 9 | 6° 2° | 70° 8° | 696 |
| 10 | 6° 30° | 72° 34° | 772 |
| 11 | 6° 34° | 75° 13° | 857 |
| 12 | 7° 41° | 77° 32° | 1097 |
| 13 | 8° 39° | 79° 6° | 1316 |
| 14 | 8° 46° | 79° 34° | 1361 |
| Pole | Vert. ∠'s at L. | Bearings at L. | Diff. of Alt. |
|------|----------------|----------------|---------------|
| 1 | 17° 46° D | 10° 52° | 226 D |
| 2 | 17° 2° | 13° 33° | 296 |
| 3 | 14° 51° | 16° 2° | 333 |
| 4 | 10° 47° | 20° 13° | 363 |
| 5 | 9° 26° | 23° 0° | 418 |
| 6 | 8° 54° | 24° 37° | 466 |
| 7 | 8° 7° | 26° 16° | 508 |
| Pole | Vert. ∠'s at T. | Bearings at e. | Diff. of Alt. |
|------|----------------|----------------|---------------|
| 1 | 11° 56° D | 16° 58° | 83 D |
| 2 | 9° 12° | 35° 25° | 120 |
| 3 | 8° 14° | 64° 20° | 190 |
| 4 | 8° 12° | 80° 0° | 250 |
| 5 | 5° 55° | 92° 52° | 234 |
| 6 | 4° 14° | 98° 36° | 193 |
| 7 | 1° 58° | 103° 12° | 101 |
| 8 | 0° 34° E | 106° 12° | 39 |
| 9 | 1° 20° | 108° 26° | 92 |
| 10 | 2° 15° | 109° 6° | 157 |
| 11 | 3° 7° | 110° 35° | 230 |
| 12 | 3° 49½° | 111° 48° | 300 |
The mean Density of the Earth.
Mr. Hutton's Calculations to ascertain
| Pole | Vert. ∠'s at F' | Bearings at F' | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 14° 37' D | 3° 57' | 73° D |
| 2 | 2° 56' | 44° 8' | 140 |
| 3 | 1° 43' | 48° 47' | 90 |
| 4 | 1° 3' | 55° 5' | 61 |
| 5 | 1° 0' | 60° 58' | 65 |
| 6 | 0° 41' | 67° 26' | 50 |
| 7 | 0° 23' | 74° 28' | 31 |
| 8 | 0° 25' E | 82° 0' | 52° A |
| 9 | 1° 53' | 83° 46' | 225 |
| 10 | 3° 17' | 87° 0' | 423 |
| 11 | 3° 32' | 87° 28' | 461 |
**SECTION 51**
| Pole | Vert. ∠'s at F' | Bearings at F' | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 3° 11' E | 64° 3' | 208° A |
| 2 | 5° 9' | 71° 24' | 366 |
| 3 | 5° 58' | 75° 50' | 448 |
| 4 | 6° 14' | 84° 12' | 522 |
| 5 | 8° 51' | 91° 23' | 821 |
| 6 | 9° 0' | 97° 47' | 924 |
| 7 | 10° 7' | 99° 40' | 1074 |
| 8 | 11° 2' | 102° 33' | 1236 |
| 9 | 11° 45' | 104° 44' | 1375 |
| 10 | 12° 6' | 106° 28' | 1469 |
| 11 | 12° 43' | 108° 24' | 1612 |
| 12 | 12° 55' | 109° 18' | 1673 |
**SECTION 52**
| Pole | Vert. ∠'s at F' | Bearings at F' | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 2° 48' E | 68° 41' | 187° A |
| 2 | 3° 50' | 83° 0' | 282 |
| 3 | 6° 45' | 102° 0' | 630 |
**SECTION 53**
### SECTION 54
| Pole | Vert. ∠'s at t' | Bearings at F' | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 0° 22' E | 14° 7 | 12 A |
| 2 | 1° 27 | 20° 5 | 44 |
| 3 | 5° 13 | 39° 2 | 241 |
| 4 | 5° 46 | 45° 18 | 299 |
| 5 | 6° 9 | 55° 40 | 379 |
| 6 | 6° 38 | 62° 0 | 451 |
| 7 | 7° 11 | 70° 8 | 553 |
| 8 | 7° 30 | 78° 56 | 662 |
| 9 | 7° 48 | 84° 30 | 755 |
| 10 | 7° 50 | 94° 8 | 908 |
| 11 | 7° 54 | 98° 26 | 1006 |
| 12 | 7° 58 | 100° 58 | 1077 |
### SECTION 56
| Pole | Vert. ∠'s at t' | Bearings at F' | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 0° 31' E | 9° 4 | 41 A |
| 2 | 4° 27 | 16° 21 | 97 |
| 3 | 7° 15 | 21° 33 | 205 |
| 4 | 9° 12 | 27° 0 | 331 |
| 5 | 10° 30 | 29° 15 | 414 |
| 6 | 11° 18 | 31° 14 | 480 |
| 7 | 11° 26 | 35° 39 | 569 |
| 8 | 12° 21 | 38° 50 | 686 |
| 9 | 12° 42 | 41° 44 | 777 |
| 10 | 13° 18 | 44° 21 | 887 |
| 11 | 13° 32 | 46° 36 | 971 |
### SECTION 57
| Pole | Vert. ∠'s at k | Bearings at t' | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 7° 51' E | 29° 43 | 176 A |
| 2 | 11° 24 | 34° 13 | 299 |
| 3 | 13° 22 | 37° 54 | 397 |
| 4 | 14° 52 | 42° 20 | 509 |
| 5 | 16° 40 | 47° 44 | 675 |
| 6 | 16° 46 | 50° 15 | 833 |
### SECTION 58
| Pole | Vert. ∠'s at k | Bearings at t' | Diff. of Alt. |
|------|----------------|---------------|--------------|
| 1 | 0° 3' E | 5° 28 | 5 A |
| 2 | 6° 10 | 8° 5 | 44 |
The following are the irregular sections. In the first column is the number of poles; in the second the vertical angles; in the third and fourth the two bearings or horizontal angles at each end of the base; and in the fifth the computed result, being the difference of altitude between the foot of each pole and the point mentioned in the second column where the vertical angles were taken.
| Pole | Vert. ∠'s at M. | Bearings at h. | Diff.of Alt. |
|------|----------------|----------------|--------------|
| 1 | 28° 16' D | Not seen. | |
| 2 | 25° 20' | 66° 56' | 910° D |
| 3 | 22° 23' | 74° 50' | 1056 |
| 4 | 22° 12' | 76° 25' | 1114 |
| 5 | 21° 40' | 79° 10' | 1232 |
| Pole | Vert. ∠'s at H. | Bear. at H. | Bear. at G. | Diff.of Alt. |
|------|-----------------|-------------|-------------|--------------|
| 1 | 8° 30'E | 70° 55' | 66° 39' | 832A |
| 2 | 8° 30' | 66° 46' | 70° 33' | 850 |
| 3 | 8° 30' | 62° 56' | 75° 9' | 884 |
| 4 | 8° 30' | 58° 20' | 79° 20' | 892 |
| 5 | 8° 30' | 54° 27' | 82° 42' | 891 |
| Pole | Vert. ∠'s at H. | Bear. at H. | Bear. at G. | Diff.of Alt. |
|------|-----------------|-------------|-------------|--------------|
| 1 | 9° 31'E | 68° 51' | 71° 0' | 1004A |
| 2 | 9° 31' | 64° 19' | 77° 50' | 1091 |
| 3 | 9° 31' | 60° 19' | 80° 38' | 1072 |
| 4 | 9° 31' | 55° 4C | 84° 34' | 1066 |
### SECTION 62
| Pole | Vert. ∠'s at H. | Bear. at H. | Bear. at G. | Diff. of Alt. |
|------|----------------|-------------|-------------|---------------|
| 1 | 11° 08' | 72° 45' | 72° 40' | 1334A |
| 2 | 11° 0 | 69° 57' | 76° 21' | 1389 |
| 3 | 11° 0 | 62° 40' | 82° 29' | 1376 |
| 4 | 11° 0 | 59° 7 | 84° 22' | 1327 |
### SECTION 63
| Pole | Vert. ∠'s at H. | Bear. at H. | Bear. at G. | Diff. of Alt. |
|------|----------------|-------------|-------------|---------------|
| 1 | 12° 15E | 74° 9 | 74° 6 | 1613A |
| 2 | 12° 15 | 70° 14 | 78° 7 | 1652 |
| 3 | 12° 15 | 67° 32 | 80° 52 | 1669 |
| 4 | 12° 15 | 64° 2 | 84° 1 | 1664 |
### SECTION 64
| Pole | Vert. ∠'s at W. | Bear. at W. | Bear. at L. | Diff. of Alt. |
|------|-----------------|-------------|-------------|---------------|
| 1 | 5° 57E | 74° 37' | 81° 12' | 382A |
| 2 | 6° 57 | 78° 12' | 76° 36' | 333 |
| 3 | 7° 11 | 81° 12' | 73° 19' | 329 |
| 4 | 6° 19 | 86° 33' | 68° 34' | 394 |
| 5 | 7° 5 | 92° 16' | 64° 4 | 334 |
| 6 | 7° 48 | 96° 43' | 60° 49' | 271 |
| 7 | 8° 40 | 100° 10' | 58° 40' | 184 |
| 8 | 8° 52 | 102° 42' | 56° 41' | 167 |
| 9 | 8° 41 | 106° 0 | 54° 23' | 168 |
| 10 | 8° 30 | 108° 53' | 52° 16' | 176 |
### SECTION 65
| Pole | Vert. ∠'s at b. | Bear. at b. | Bear. at a. | Diff. of Alt. |
|------|-----------------|-------------|-------------|---------------|
| 1 | 2° 1D | 47° 47' | 19° 56' | 20D |
| 2 | 1° 15E | 32° 1 | 26° 45' | 26A |
| 3 | 2° 59 | 27° 0 | 32° 56' | 65 |
| 4 | 4° 35 | 23° 40' | 45° 18' | 118 |
| 5 | 5° 0 | 19° 2 | 63° 47' | 150 |
| 6 | 5° 39 | 15° 26' | 98° 12' | 202 |
| 7 | 6° 1 | 13° 3 | 121° 47' | 238 |
| 8 | 5° 44 | 9° 29' | 143° 12' | 247 |
### SECTION 66
| Pole | Vert. ∠'s at d. | Bear. at d. | Bear. at c. | Diff. of Alt. |
|------|-----------------|-------------|-------------|---------------|
| 1 | 14° 25D | 162° 37' | 11° 30' | 870D |
| 2 | 15° 55 | 158° 15' | 13° 36' | 823 |
| 3 | 16° 57 | 152° 52' | 16° 35' | 827 |
| 4 | 18° 12 | 149° 43' | 18° 0 | 831 |
| 5 | 20° 57 | 147° 3 | 18° 41' | 866 |
| 6 | 23° 5 | 136° 18' | 23° 37' | 865 |
| 7 | 23° 40 | 129° 0 | 27° 23' | 876 |
| 8 | 23° 3 | 122° 16' | 31° 32' | 877 |
| 9 | 23° 46 | 112° 59' | 36° 25' | 894 |
| 10 | 23° 23 | 104° 22' | 41° 34' | 891 |
| 11 | 23° 35 | 91° 13' | 48° 53' | 892 |
| 12 | 21° 21 | 82° 49' | 57° 38' | 902 |
| 13 | 20° 11 | 77° 33' | 63° 0 | 897 |
| 14 | 18° 47 | 69° 33' | 71° 24' | 890 |
| 15 | 17° 31 | 65° 26' | 77° 38' | 893 |
Mr. Hutton's Calculations to ascertain
| Pole | Vert. ∠'s at P. | Bear. at P. | Bear. at H. | Diff.of Alt. |
|------|----------------|-------------|-------------|--------------|
| 1 | 5 42D | 47 54 | 61 12 | 476D |
| 2 | 7 42 | 54 10 | 53 56 | 592 |
| 3 | 7 13 | 55 23 | 52 22 | 542 |
| 4 | 7 12 | 58 40 | 50 36 | 532 |
| 5 | 8 1 | 63 52 | 46 43 | 564 |
| 6 | 6 47 | 68 29 | 44 48 | 469 |
| 7 | 5 27 | 71 25 | 43 18 | 369 |
| 8 | 5 31 | 79 14 | 40 16 | 368 |
| 9 | 4 15 | 81 17 | 39 11 | 278 |
| 10 | 3 51 | 87 37 | 36 34 | 247 |
| 11 | 3 48 | 90 58 | 35 16 | 242 |
| 12 | 1 5 | 96 47 | 32 56 | 64 |
| 13 | 1 27 | 96 53 | 30 46 | 80 |
| 14 | 3 42 | 91 16 | 32 0 | 208 |
| 15 | 6 0 | 83 54 | 34 35 | 348 |
| 16 | 7 21 | 69 39 | 38 26 | 435 |
| 17 | 9 5 | 64 1 | 41 18 | 564 |
| 18 | 9 34 | 55 56 | 44 56 | 625 |
| 19 | 10 1 | 48 45 | 49 59 | 706 |
| 20 | 9 24 | 41 45 | 54 41 | 703 |
| 21 | 9 52 | 37 19 | 59 16 | 777 |
SECTION 68.
| Pole | Vert. ∠'s at P. | Bear. at P. | Bear. at G. | Diff.of Alt. |
|------|----------------|-------------|-------------|--------------|
| 1 | 6 23D | 21 51 | | |
| 2 | 6 20 | 22 44 | 139 42 | 1140D |
| 3 | 6 20 | 24 2 | 136 17 | 1092 |
| 4 | 6 14 | 25 20 | 132 35 | 1025 |
| 5 | 5 54 | 27 20 | 127 52 | 932 |
| 6 | 5 41 | 29 23 | 121 57 | 843 |
| 7 | 5 21 | 32 27 | 115 58 | 769 |
| 8 | 5 29 | 35 59 | 107 50 | 740 |
| 9 | 5 45 | 41 31 | 87 8 | 615 |
| 10 | 5 30 | 45 10 | 82 44 | 578 |
| 11 | 5 42 | 49 50 | 77 53 | 589 |
| 12 | 5 56 | 53 57 | 74 41 | 613 |
| 13 | 6 3 | 58 6 | 70 2 | 605 |
| 14 | 5 4 | 64 0 | 66 29 | 510 |
| 15 | 5 8 | 67 27 | 65 5 | 527 |
| 16 | 4 11 | 73 46 | 65 27 | 486 |
| 17 | 4 20 | 76 52 | 64 11 | 518 |
| 18 | 4 21 | 80 26 | 62 58 | 542 |
SECTION 69.
| Pole | Vert. ∠'s at H. | Bear. at H. | Bear. at W. | Diff.of Alt. |
|------|----------------|-------------|-------------|--------------|
| 1 | 12 47E | 116 58 | 51 34 | 1809A |
| 2 | 14 36 | 107 42 | 59 25 | 2135 |
The three following sections were taken in a manner different from all the rest. They were made by measuring in a straight sloping line (or nearly straight) from certain points towards K and N, and at the beginning of the line taking the angle of elevation or depression of several places or points in it, whose distance from the beginning were measured. In these cases each distance is the hypotenuse of a right-angled triangle, and the manner of operation is this, as radius is to the hypotenuse or measured slope distance, so is the sine of the elevation or depression to the difference of altitude, and so is the cosine of the same vertical angle to the horizontal distance.
| Pole | Slope Dist. | Vert. ∠'s at M' | Horiz. Dist. | Diff.of Alt. |
|------|-------------|----------------|--------------|--------------|
| 1 | 463 | 7° 50' 2" | 459 | 60 |
| 2 | 794 | 6° 54' 2" | 788 | 92 |
| 3 | 992 | 6° 50' | 985 | 114 |
SECTION 70, from M' to K.
| Pole | Slope Dist. | Vert. ∠'s at M' | Horiz. Dist. | Diff.of Alt. |
|------|-------------|----------------|--------------|--------------|
| 4 | 1257 | 5° 28' 2" | 1251 | 116 |
| 5 | 1455 | 5° 25' 2" | 1449 | 134 |
| 6 | 1824 | 5° 4 | 1817 | 157 |
ENDS at K.
Besides these sections there were many more single points, whose places and relative altitudes were observed and computed, but it is not necessary to abstract them all here.
The following plate (Tab. viii.) has 72 figures answering to these 72 sections, each to each, according to the numbers. In these figures, the line having the letters p, p, p, &c. annexed is the section line, the letters p, p, &c. denoting the poles; the other line, forming the angle with the section, is the base line; and between them are the degrees and minutes contained in the angle formed by them; at the angular point was observed the elevation or depression of each point p, and the bearings or horizontal angles were observed at the other end of the base, from whence faint lines are drawn to some of the points p forming with the base line those horizontal angles.
angles. The base and section lines in each figure are also drawn nearly in the same direction as they are in the plan or on the ground, supposing the top of the paper to be the North, towards which a person looks when viewing the ground from the South.
Having finished the computation of the relative altitudes of all the points, the next consideration is how they are to be applied in determining the attraction of the hill. In whatever manner this last mentioned operation may be performed, it is evident, that all the points and sections with their altitudes must be entered in the plan. Wherefore, having accurately constructed a large plan of the ground, as before mentioned, containing all the principal lines or bases, at the extremities of which either vertical or horizontal angles were taken, from them I then determined in this plan the places of all the other points in the sections, whether vertical, horizontal, or irregular. These places or points were determined by drawing lines from each extremity of the base so as to form with it angles equal to those which were observed on the ground for each corresponding pole; the intersections of these lines are the places of the poles, which having marked with a fine dot or point of ink, and written close to each point the proper number expressing its relative altitude, and cleaned the paper by rubbing out the lines forming the
angles by which the points were determined, there remained only the points with the figures expressing their altitudes distinctly exhibited in the plan (see tab. ix.).
It remains now to apply all the foregoing calculations and constructions to the determination of the effect of the attraction in the direction of the meridian. And here it soon occurred, that the best method was to divide the plan into a great number of small parts, which may be considered as the bases of as many vertical columns or pillars of matter into which the hill and the adjacent ground may be supposed to be divided by vertical planes, forming an imaginary group of vertical columns, something like a set of basaltine pillars, or like the cells in a piece of honey-comb; then to compute the attraction of each pillar separately in the direction of the meridian; and lastly, to take the sum of all these computed effects for the whole attraction of the matter in the hill, &c.
Now the attraction of any one of these pillars on a body in a given place may be easily determined, and that in any direction, to a sufficient degree of accuracy, because of the smallness and given position of the base; for, on account of its smallness, all the matter in the pillar may be supposed to be collected into its axis or vertical line erected on the middle of the base, the length of which axis, as the mean altitude of the pillar, is to be estimated from
from the altitudes of the points in the plan which fall within and near the base of the pillar: then, having given the altitude of this axis, with the position of its base, and the matter supposed to be collected into it, a theorem can easily be given by which the effect of its attraction may be computed. But to retain the proper degree of accuracy in this computation, it is evident that the plan must be divided into a great number of parts, perhaps not less than a thousand for each observatory, in order that they may be sufficiently small, and by this means forming about two thousand of such pillars of matter, whose attractions must be separately computed, as mentioned above. The labour and time necessary for such computation, it is evident, would be very great, perhaps not less than those employed in all the preceding computations of the sections, and all the other points and lines concerned in this business. For this reason I was desirous of obtaining a theorem or method by which the attractions of the small and numerous pillars might be computed with the same degree of accuracy, but with less expense of labour and time than when computed separately as above mentioned. And in this inquiry the success has been equal to my wishes, having at length met with a method by which the business has been effected in perhaps one-fourth or one-fifth of the time
Mr. Hutton's Calculations to ascertain time that would have been required in the other way. This method I have investigated partly from some hints of the honourable Henry Cavendish, F.R.S. and partly from some of my own, which had been communicated to the Astronomer Royal in the years 1774 and 1775: of which method and its investigation I shall now give some account.
Of all the methods of dividing the plan into a great number of small parts, I have found that to be the most convenient for the computation, in which it is first divided into a number of rings by concentric circles, and these again divided into a sufficient number of parts by radii drawn from the common center, that center being the observatory where the plummet is placed on which the effect of attraction is to be computed. By this means the plan is divided into a great number of small quadrilateral spaces, two of the opposite sides of which are small portions of adjacent circles, and the other two are the intercepted small parts of two adjacent radii, as appears by fig. 1. tab. x. in which, for the present, let the circles and their radii be supposed to be drawn at any distances whatever from each other, till it shall appear from the theorem to be investigated what may be the properest distances and positions of those lines. In this figure A is the observatory, AN the meridian, NAE an
East-and-West line, BCDE one of the little spaces, and F its center or foot of the axis of the pillar whose base is BCDE; the figure AWNEA being a horizontal or level section through the point A. Join A, F, and with the center A describe the middle circle GH. Let \(a\) denote the length of the axis on the point F, or the mean height of the pillar on the base BD; and \(s = \frac{a}{\sqrt{a^2 + AF^2}}\). Then will the magnitude of that column or its quantity of matter be expressed by \(\frac{BC + ED}{2} \times BE \times a\), which is supposed to be all collected into the axis: consequently, if the attraction of each particle of matter be in the reciprocal duplicate ratio of its distance, the attraction of the matter in the pillar, so placed on the plummet at A, in the direction of the meridian AN, will be
\[
\frac{BC + ED}{2AF} \times BE \times a \times \frac{s}{a} \times c = \frac{BC + ED}{2AF} \times BE \times SC = \frac{GH}{AF} \times BE \times SC
\]
nearly, supposing F to be equally distant from BC and ED, and \(c\) the cosine of the angle FAN to the radius 1.
But \(\frac{GH}{AF} \times c\) is nearly equal to \(d\) the difference of the sines of the angles BAN, CAN, as is thus demonstrated. Draw GK, FL, HM, perpendicular, and GP parallel to AW; and draw the chord GH. Then AK, AM are the sines of the angles GAN, HAN, to the radius AF, their difference being
being $KM = GP$; also $FL$ is the cosine of $FAN$ to the same radius: consequently $GP : FL = d : c$. But the triangles $LFA$, $PGH$ are equiangular, and therefore $GP : FL = GH : AF$. Consequently $GH : AF = d : c$; or $\frac{GH}{AF} \times c = d$. This equation is accurately true when $GH$ is the chord of the arc; and as the small arc differs insensibly from its chord, the same equation is sufficiently near the truth when $GH$ is the arc itself. Substituting now $d$ instead of the quantity $\frac{GH}{AF} \times c$ in the theorem above, it will become $BE \times ds$ for the measure of the attraction of the pillar whose base is $BD$ in the direction $AN$. Which is as easy and simple an expression for the attraction of a single pillar as can well be desired or expected.
But to make the application of this theorem still more easy to the great number of small pillars concerned in this business, let us suppose $BE$ and $d$ to be constant or invariable quantities, and then it is evident that we shall have nothing more to do but to collect all the $s$'s or fines of elevation of all the pillars into one sum, and then multiply that sum by the constant quantity $BE \times d$, by which there will be produced the measure of the attraction of all the pillars, or of the whole part of the ground on one side of $WE$. Now $BE$ will be made to become constant, by making the circles equi-distant from one another,
another, or by taking the radii in arithmetical progression. And \( d \) will be constant, by drawing the radii so as to form with angles whose sines shall be in arithmetical progression; for then \( d \) is the common difference of the sines of those angles. Hence then we are easily led to the best manner of dividing the plan into the small spaces, viz. from the center \( A \) describe a sufficient number of concentric and equi-distant circles; divide the radius \( AI \) of any one of them into a sufficient number of equal parts, and from the points of division erect perpendiculars to meet the circle; then through the points of intersection draw radii, and they will divide the circles in the manner required.
In a computation of this kind, we need only calculate the attraction of the matter above the plane or horizon of each observatory, and the attraction of so much matter as is wanting to fill up the vacuity below that plane lying between it and the surface of the lower part of the hill. For the South observatory, the attraction of the Southern parts that are above it must be subtracted from that of the Northern parts, to obtain the attraction of the whole towards the North; that is, the Southern elevations are negative, and the Northern ones affirmative. The contrary names take place with respect to the depressions, or the vacuities below the plane of the observatory; for if
Mr. Hutton's Calculations to ascertain
the whole space below this horizontal plane were full of matter to an equal extent both ways, its attraction need not be computed, as those on the contrary sides would mutually balance each other; but since there are unequal vacuities on each side, it is evident, that the attraction of the matter that might be contained in them must be deducted from the other two equal quantities, to leave the real attraction of those two sides; then subtracting the remainder to the South side from that of the Northern side, there will at last remain the joint effect of all the matter below the plane in the Northern direction: but as the one remainder is to be subtracted from the other, the two equal quantities may be omitted in both, and only the effects of the vacuities brought into the account, which being twice subtracted, their signs become contrary to those of the parts above the horizontal plane; that is, the effect of the Southern vacuity is affirmative, and that of the Northern one negative. But for the Northern observatory, when the attraction towards the South is to be found, the contrary names take place; that is, in the elevations the Southern parts are affirmative, and the Northern parts negative; but in the vacuities or depressions, the Northern parts are affirmative, and the Southern ones negative.
According
According to the foregoing method the plan of the ground was divided into 20 rings by equidistant concentric circles, described about each observatory as a center; and each quadrant was divided into 12 parts or sectors by lines forming, with the meridian, angles whose sines are in arithmetical progression; by which means the space in each quadrant was divided into 240 small parts, making almost a thousand of such parts in the whole round for each observatory, or near 2000 for the two observatories. This was judged to be a sufficiently great number of parts to afford a very considerable degree of accuracy; or at least that number was as great, and the parts as small, as was well consistent with the degree of accuracy afforded by the number of points whose relative altitudes had been determined.
In this division the common breadth of the rings, or the common difference of the radii, is $666\frac{2}{3}$ feet; and the common difference of the sines of the angles formed by the radii and the meridian is $\frac{1}{12}$th of the radius; and consequently, those angles are expressed in degrees and minutes as here follows, viz. $4^\circ 47'$, $9^\circ 36'$, $14^\circ 29'$, $19^\circ 28'$, $24^\circ 37'$, $30^\circ 0'$, $35^\circ 41'$, $41^\circ 48'\frac{1}{2}$, $48^\circ 35'$, $56^\circ 26'\frac{1}{2}$, $66^\circ 26'\frac{1}{2}$, $90^\circ 0'$.
Tab. IX. contains a small plan of the principal and most central part of the ground, accurately divided in the above
Mr. Hutton's Calculations to ascertain above manner for one of the observatories, namely, the Northern one, with the places of all or most of the points which fall within this part of the ground, accurately laid down and marked with dots, as also such of the included letters as have been before mentioned in this paper.
In this plate \( \text{RABCD} \), &c. is the chain of stations around the hill; \( N \) and \( K \) are the West and East cairns on the extremities of the ridge of the hill; \( O \) the Southern observatory, and \( P \) the Northern one. Of this kind were made two large plans, one divided for each observatory, from which were estimated the mean altitudes of the pillars erected on the spaces into which they are divided.
These altitudes are easily estimated when several of the points fall near and in the small spaces or bases, especially when they are near the middle of them; but, numerous as the points are, there are evidently many bases in which none at all are contained, nor even near them. This circumstance at first gave me much trouble and dissatisfaction, till I fell upon the following method by which the defect was in a great measure supplied, and by which I was enabled to proceed in the estimation of the altitudes both with much expedition and a considerable degree of accuracy. This method was the connecting together by a faint line all the points which were
of the same relative altitude: by so doing, I obtained a great number of irregular polygons lying within, and at some distance from, one another, and bearing a considerable degree of resemblance to each other: these polygons were the figures of so many level or horizontal sections of the hills, the relative altitudes of all the parts of them being known; and as every base or little space had several of them passing through it, I was thereby able to determine the altitude belonging to each space with much ease and accuracy. In this estimation I could generally be pretty sure of the altitude to within ten feet, and often within five, which on an average might be about the 100th part of the whole altitude; and when we consider that the number of such estimated altitudes is very great, and that it is probable the small errors among them would nearly balance one another, the defect of those that might be reckoned too little being compensated by the excess in those which might be taken too great, we need not hesitate to pronounce, that the error arising from the estimation of the altitudes is probably still much less than that part.
It was necessary to determine these altitudes of the pillars, in order to compute the sines of the angles of elevation subtended by them, as the theorem requires the use of these sines; and the very easy method used in
deducing the latter from the former shall be explained after we have, as below, registered the altitudes of all the pillars as they were computed. This register consists of sixteen tables, namely four quadrants of spaces in the altitudes, and four in the depressions, for each observatory, as specified in the titles of them. The numbers are feet, like all the other dimensions. The numbers on the same horizontal line from left to right are such as are all in the same ring; and those in one and the same vertical column are in the same sector, or between the same two radii; the number of the ring, counted from the common center, is written in the left-hand margin; and the number of the vertical column or distance of the space or sector from the meridian, at the top; also the radius of each ring, that is, the line from the common center to the middle of the ring is written on the same line with it, in the right-hand margin. It may be further remarked, that in such little spaces as were cut through by the boundary line between elevations and depressions, thereby making but a part of such spaces in each of those denominations, each space was accounted as a whole one; but then the mean altitude or depression in each part was diminished in the proportion of the whole space to the part of it so included in the boundary. The altitudes and depressions are put down first with respect to
the Southern observatory o, and then for the Northern observatory p; and in each, the altitudes are placed first.
1. Altitudes above 0 in the N.W. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| | 215 | 215 | 215 | 215 | 210 | 205 | 200 | 190 | 170 | 145 | 120 | 75 |
| 1 | 605 | 610 | 605 | 600 | 595 | 590 | 580 | 570 | 510 | 450 | 350 | 200 |
| 2 | 965 | 1005| 1010| 1010| 1020| 1050| 1040| 960 | 810 | 600 | 415 | 220 |
| 3 | 670 | 680 | 700 | 780 | 860 | 930 | 1040| 1090| 700 | 480 | 210 | 2333|
| 4 | 280 | 310 | 370 | 450 | 560 | 700 | 830 | 960 | 1180| 890 | 545 | 200 |
| 5 | 20 | 50 | 100 | 110 | 250 | 380 | 525 | 710 | 890 | 950 | 605 | 110 |
| 6 | | | | | | | | | | | | |
| 7 | | | | | | | | | | | | |
| 8 | | | | | | | | | | | | |
| 9 | | | | | | | | | | | | |
| 10 | | | | | | | | | | | | |
2. Altitudes above 0 in the N.E. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| | 210 | 205 | 205 | 200 | 195 | 185 | 170 | 155 | 140 | 125 | 105 | 70 |
| 1 | 550 | 545 | 540 | 530 | 520 | 510 | 500 | 465 | 430 | 370 | 270 | 130 |
| 2 | 910 | 840 | 825 | 815 | 800 | 760 | 720 | 680 | 635 | 590 | 500 | 200 |
| 3 | 645 | 640 | 635 | 640 | 645 | 650 | 675 | 715 | 730 | 700 | 580 | 300 |
| 4 | 265 | 255 | 265 | 285 | 310 | 350 | 390 | 450 | 460 | 500 | 600 | 280 |
| 5 | 10 | 12 | 20 | 65 | 100 | 130 | 160 | 180 | 180 | 320 | 460 | 300 |
| 6 | | | | | | | | | | | | |
| 7 | | | | | | | | | | | | |
| 8 | | | | | | | | | | | | |
3. Altitudes above 0 in the S.W. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| | | | | | | | | | | | | |
| 1 | | | | | | | | | | | | |
| 2 | | | | | | | | | | | | |
| 3 | | | | | | | | | | | | |
| 4 | | | | | | | | | | | | |
| 5 | | | | | | | | | | | | |
| 6 | | | | | | | | | | | | |
| 7 | | | | | | | | | | | | |
| 8 | | | | | | | | | | | | |
| 9 | | | | | | | | | | | | |
| 10 | | | | | | | | | | | | |
| 11 | | | | | | | | | | | | |
| 12 | | | | | | | | | | | | |
| 13 | | | | | | | | | | | | |
| 14 | | | | | | | | | | | | |
| 15 | | | | | | | | | | | | |
| 16 | | | | | | | | | | | | |
| 17 | | | | | | | | | | | | |
| 18 | | | | | | | | | | | | |
| 19 | | | | | | | | | | | | |
| 20 | | | | | | | | | | | | |
### 4. Altitudes above 0 in the S.E. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Radii |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-------|
| 16 | 10 | | | | | | | | 10333 |
| 17 | 80 | 60 | 50 | 50 | 10 | | | | 11000 |
| 18 | 220 | 200 | 180 | 130 | 70 | 30 | | | 11667 |
| 19 | 340 | 300 | 260 | 240 | 170 | 120 | 20 | | 12333 |
| 20 | 450 | 410 | 380 | 330 | 260 | 180 | 100 | 20 | 13000 |
### 5. Depressions below 0 in the N.W. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Radii |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-------|
| 6 | 70 | 40 | 15 | 5 | | | | | | | | | 15 |
| 7 | 250 | 240 | 200 | 150 | 60 | 30 | | | | | | | 40 |
| 8 | 460 | 450 | 430 | 390 | 280 | 200 | 80 | 10 | | | | | 80 |
| 9 | 700 | 700 | 680 | 630 | 520 | 450 | 340 | 170 | 15 | | | | 120 |
| 10 | 840 | 830 | 800 | 780 | 650 | 600 | 520 | 380 | 180 | 40 | 70 | 220 | 6333 |
| 11 | 960 | 920 | 880 | 850 | 750 | 650 | 630 | 550 | 350 | 250 | 300 | 430 | 7000 |
| 12 | 1100| 1000| 950 | 900 | 820 | 780 | 780 | 580 | 530 | 500 | 560 | 667 | 7667 |
| 13 | 1130| 1080| 980 | 880 | 840 | 800 | 830 | 860 | 780 | 690 | 630 | 640 | 8333 |
| 14 | 1180| 1100| 1000| 900 | 900 | 910 | 940 | 870 | 800 | 700 | 500 | | 9000 |
| 15 | 1180| 1100| 1100| 1080| 1040| 1050| 1060| 1070| 1000| 870 | 730 | 300 | 9667 |
| 16 | 1100| 1100| 1100| 1100| 1100| 1140| 1150| 1150| 1120| 990 | 760 | 160 | 10333|
| 17 | 1100| 1100| 1100| 1130| 1180| 1200| 1200| 1200| 1180| 1080| 700 | 80 | 11000|
| 18 | 1100| 1100| 1150| 1200| 1200| 1150| 1100| 1100| 1200| 1180| 700 | 100 | 11667|
| 19 | 1100| 1120| 1220| 1230| 1260| 1200| 1200| 1200| 1300| 1240| 620 | 60 | 12333|
| 20 | 1120| 1220| 1320| 1360| 1390| 1390| 1390| 1340| 1440| 1300| 620 | 50 | 13000|
### 6. Depressions below 0 in the N.E. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Radii |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-------|
| 6 | 70 | 60 | 30 | 10 | | | | | | | | | 10 |
| 7 | 260 | 240 | 200 | 150 | 110 | 80 | 30 | 40 | 30 | 10 | | | 40 |
| 8 | 450 | 440 | 400 | 350 | 280 | 180 | 100 | 180 | 180 | 190 | 190 | 40 | 5000 |
| 9 | 700 | 690 | 680 | 610 | 520 | 400 | 200 | 240 | 290 | 350 | 330 | 200 | 5667 |
| 10 | 850 | 870 | 890 | 860 | 770 | 620 | 440 | 300 | 380 | 500 | 450 | 370 | 6333 |
| 11 | 1020| 1060| 1070| 1050| 980 | 860 | 700 | 520 | 400 | 650 | 600 | 530 | 7000 |
| 12 | 1140| 1160| 1180| 1160| 1140| 1080| 950 | 840 | 620 | 720 | 850 | 700 | 7667 |
| 13 | 1200| 1190| 1200| 1220| 1240| 1250| 1160| 1050| 900 | 840 | 950 | 880 | 8333 |
| 14 | 1230| 1310| 1050| 1050| 1100| 1220| 1260| 1220| 1070| 950 | 1020| 990 | 9000 |
| 15 | 1100| 960 | 900 | 850 | 900 | 1100| 1230| 1210| 1170| 1060| 1090| 1100| 9667 |
| 16 | 970 | 860 | 880 | 780 | 780 | 900 | 1120| 1180| 1200| 1180| 1160| 1150| 10333|
| 17 | 970 | 800 | 760 | 750 | 750 | 780 | 1000| 1200| 1300| 1240| 1200| 1100| 11000|
### 7. Depressions below 0 in the S.W. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Radii |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-------|
| 1 | 165 | 165 | 160 | 155 | 150 | 140 | 130 | 120 | 110 | 90 | 50 | 10 | 333 |
| 2 | 400 | 390 | 380 | 370 | 350 | 330 | 300 | 270 | 240 | 210 | 160 | 60 | 1000 |
| 3 | 600 | 580 | 560 | 530 | 500 | 570 | 540 | 400 | 370 | 340 | 280 | 100 | 1667 |
| 4 | 740 | 720 | 700 | 670 | 640 | 610 | 580 | 530 | 490 | 440 | 370 | 160 | 2333 |
| 5 | 800 | 800 | 800 | 770 | 740 | 710 | 660 | 610 | 570 | 510 | 440 | 230 | 3000 |
| 6 | 780 | 790 | 780 | 770 | 780 | 800 | 790 | 700 | 650 | 590 | 510 | 320 | 3667 |
| 7 | 700 | 710 | 720 | 730 | 750 | 750 | 750 | 750 | 730 | 670 | 600 | 400 | 4333 |
| 8 | 580 | 590 | 600 | 610 | 640 | 660 | 700 | 720 | 730 | 730 | 760 | 520 | 5000 |
| 9 | 490 | 490 | 490 | 480 | 490 | 510 | 600 | 650 | 660 | 690 | 580 | 450 | 5667 |
| 10 | 470 | 460 | 420 | 400 | 420 | 420 | 440 | 490 | 580 | 590 | 560 | 430 | 6333 |
| 11 | 340 | 340 | 340 | 340 | 340 | 330 | 350 | 390 | 450 | 480 | 380 | 370 | 7000 |
| 12 | 210 | 220 | 230 | 250 | 250 | 250 | 280 | 310 | 340 | 370 | 250 | 200 | 7067 |
| 13 | 160 | 150 | 140 | 120 | 130 | 150 | 200 | 230 | 280 | 290 | 230 | 110 | 8333 |
| 14 | 110 | 90 | 60 | 20 | 20 | 20 | 70 | 150 | 230 | 240 | 150 | 90 | 9000 |
| 15 | 50 | 20 | | | | | | | | | | | 9667 |
| 16 | | | | | | | | | | | | | 10333 |
### 8. Depressions below 0 in the S.E. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Radii |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-------|
| 1 | 165 | 165 | 160 | 155 | 150 | 140 | 130 | 120 | 110 | 100 | 95 | 40 | 333 |
| 2 | 400 | 400 | 400 | 400 | 400 | 390 | 380 | 350 | 330 | 300 | 250 | 110 | 1000 |
| 3 | 600 | 610 | 610 | 610 | 610 | 600 | 600 | 580 | 550 | 500 | 440 | 200 | 1667 |
| 4 | 760 | 750 | 740 | 740 | 740 | 730 | 720 | 710 | 680 | 640 | 560 | 300 | 2333 |
| 5 | 800 | 800 | 800 | 800 | 800 | 800 | 800 | 790 | 740 | 660 | 400 | 3000 |
| 6 | 780 | 780 | 780 | 780 | 780 | 790 | 800 | 840 | 880 | 850 | 750 | 470 | 3667 |
| 7 | 700 | 690 | 680 | 670 | 660 | 680 | 620 | 720 | 820 | 900 | 770 | 520 | 4333 |
| 8 | 580 | 570 | 570 | 570 | 570 | 580 | 590 | 600 | 660 | 800 | 800 | 600 | 5000 |
| 9 | 490 | 490 | 490 | 490 | 490 | 490 | 490 | 500 | 520 | 700 | 880 | 600 | 5667 |
| 10 | 470 | 460 | 450 | 440 | 440 | 430 | 420 | 410 | 430 | 470 | 530 | 880 | 6333 |
| 11 | 340 | 330 | 320 | 320 | 320 | 320 | 330 | 350 | 420 | 500 | 780 | 780 | 7000 |
| 12 | 210 | 200 | 200 | 200 | 210 | 220 | 240 | 280 | 390 | 480 | 680 | 900 | 7667 |
| 13 | 120 | 120 | 130 | 130 | 140 | 150 | 180 | 230 | 300 | 450 | 600 | 990 | 8333 |
| 14 | 110 | 110 | 110 | 120 | 130 | 150 | 160 | 200 | 280 | 440 | 580 | 930 | 6000 |
| 15 | 70 | 70 | 70 | 70 | 90 | 120 | 140 | 170 | 240 | 420 | 570 | 960 | 9667 |
| 16 | 10 | 20 | 30 | 40 | 50 | 80 | 120 | 190 | 220 | 400 | 550 | 1000 | 10333 |
| 17 | | | | | | | | | | | | | 1100 |
| 18 | | | | | | | | | | | | | 1166 |
| 19 | | | | | | | | | | | | | 1233 |
| 20 | | | | | | | | | | | | | 1300 |
Vol. LXVIII.
9. Altitudes above P in the N.W. quarter.
| Rings | 12 Radii |
|-------|----------|
| 4 | 2333 |
| 5 | 3000 |
| 6 | 3667 |
| 7 | 4333 |
| 8 | 5000 |
10. Altitudes above P in the N.E. quarter.
| Rings | 12 Radii |
|-------|----------|
| 1 | 3333 |
| 2 | 1000 |
| 3 | 1667 |
| 4 | 2333 |
| 5 | 3000 |
| 6 | 3667 |
11. Altitudes above P in the S.W. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| | 110 | 110 | 105 | 105 | 100 | 100 | 95 | 95 | 90 | 85 | 80 | 35 |
| 2 | 340 | 330 | 320 | 310 | 300 | 290 | 280 | 270 | 240 | 210 | 170 | 90 |
| 3 | 660 | 660 | 660 | 660 | 660 | 650 | 620 | 590 | 570 | 510 | 370 | 170 |
| 4 | 1020| 1030| 1040| 1050| 1060| 1070| 1030| 990 | 910 | 800 | 660 | 270 |
| 5 | 1020| 1110| 1280| 1270| 1320| 1330| 1310| 1280| 1270| 1170| 910 | 460 |
| 6 | 670 | 770 | 810 | 900 | 930 | 950 | 1020| 1070| 1150| 1270| 1100| 600 |
| 7 | 280 | 340 | 420 | 480 | 540 | 570 | 620 | 670 | 720 | 880 | 1050| 660 |
| 8 | 20 | 50 | 90 | 140 | 210 | 290 | 350 | 420 | 490 | 570 | 700 | 570 |
| 9 | | | | | | | | | | | | |
| 10 | | | | | | | | | | | | |
| 11 | | | | | | | | | | | | |
| 12 | | | | | | | | | | | | |
### 12. Altitudes above P in the S.E. quadrant.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| | 110 | 110 | 105 | 105 | 100 | 100 | 95 | 95 | 90 | 85 | 80 | 35 |
| 1 | 340 | 330 | 320 | 310 | 300 | 290 | 280 | 270 | 240 | 210 | 170 | 90 |
| 2 | 660 | 640 | 620 | 600 | 570 | 540 | 510 | 480 | 440 | 380 | 300 | 160 |
| 3 | 1000| 980 | 950 | 910 | 870 | 810 | 730 | 670 | 540 | 460 | 330 | 170 |
| 4 | 1020| 1020| 1020| 1030| 1020| 970 | 770 | 570 | 470 | 390 | 130 | 3000|
| 5 | 670 | 710 | 770 | 810 | 840 | 860 | 910 | 890 | 720 | 650 | 400 | 30 |
| 6 | 290 | 320 | 360 | 390 | 470 | 590 | 700 | 750 | 780 | 600 | 280 | 4333|
| 7 | 20 | 70 | 170 | 250 | 420 | 630 | 550 | 170 | 5000| 5667| |
| 8 | | | | | | | | | | | | |
| 9 | | | | | | | | | | | | |
| 10 | | | | | | | | | | | | |
| 18 | | | | | | | | | | | | |
| 19 | | | | | | | | | | | | |
| 20 | | | | | | | | | | | | |
### 13. Depressions below P in the N.W. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| | 100 | 95 | 90 | 85 | 80 | 75 | 70 | 60 | 50 | 40 | 30 | 15 |
| 1 | 390 | 380 | 360 | 330 | 310 | 290 | 270 | 240 | 210 | 180 | 150 | 60 |
| 2 | 520 | 510 | 500 | 490 | 480 | 470 | 450 | 430 | 410 | 370 | 270 | 80 |
| 3 | 650 | 640 | 620 | 610 | 590 | 570 | 550 | 530 | 500 | 460 | 390 | 90 |
| 4 | 830 | 820 | 760 | 720 | 690 | 660 | 630 | 590 | 560 | 500 | 380 | 130 |
| 5 | 880 | 860 | 850 | 790 | 730 | 700 | 670 | 640 | 580 | 480 | 340 | 280 |
| 6 | 910 | 900 | 860 | 830 | 790 | 720 | 630 | 620 | 540 | 550 | 440 | 185 |
| 7 | 930 | 890 | 840 | 800 | 830 | 710 | 610 | 610 | 580 | 530 | 520 | 430 |
| 8 | 830 | 830 | 830 | 830 | 830 | 760 | 700 | 670 | 620 | 600 | 330 | 5667|
| 9 | 730 | 740 | 755 | 770 | 785 | 800 | 815 | 830 | 780 | 750 | 720 | 460 |
| 10 | 750 | 770 | 810 | 860 | 910 | 930 | 950 | 960 | 950 | 930 | 880 | 530 |
| 11 | 770 | 840 | 910 | 950 | 990 | 1030| 1050| 1030| 980 | 950 | 950 | 650 |
| 12 | | | | | | | | | | | | |
| 13 | | | | | | | | | | | | |
### 14. Depressions below p in the N.E. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| | 80 | 75 | 70 | 65 | 60 | 55 | 50 | 45 | 40 | 35 | 30 | 20 |
| | 330 | 325 | 320 | 300 | 280 | 260 | 240 | 220 | 190 | 150 | 110 | 30 |
| | 520 | 515 | 505 | 490 | 475 | 460 | 440 | 420 | 400 | 330 | 240 | 60 |
| | 660 | 675 | 690 | 700 | 700 | 690 | 660 | 620 | 560 | 470 | 350 | 25 |
| | 840 | 840 | 840 | 840 | 830 | 820 | 770 | 720 | 620 | 440 | 100 | 300 |
| | 860 | 880 | 900 | 920 | 930 | 930 | 910 | 870 | 830 | 740 | 570 | 240 |
| | 920 | 920 | 920 | 880 | 880 | 900 | 930 | 940 | 930 | 840 | 680 | 610 |
| | 920 | 840 | 780 | 780 | 740 | 720 | 770 | 870 | 920 | 970 | 900 | 630 |
| | 720 | 670 | 600 | 600 | 600 | 560 | 580 | 670 | 850 | 950 | 940 | 600 |
| | 700 | 620 | 520 | 500 | 500 | 500 | 500 | 520 | 720 | 920 | 960 | 650 |
| | 700 | 600 | 600 | 600 | 620 | 600 | 580 | 560 | 540 | 840 | 920 | 770 |
| | 720 | 700 | 680 | 700 | 720 | 740 | 700 | 740 | 570 | 800 | 920 | 820 |
| | 720 | 720 | 720 | 720 | 700 | 700 | 720 | 720 | 620 | 820 | 900 | 920 |
### 15. Depressions below p in the S.W. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| | 230 | 140 | 100 | 70 | 40 | 40 | | | | | | |
| | 400 | 340 | 280 | 230 | 190 | 150 | 110 | 30 | | | | |
| | 500 | 470 | 410 | 340 | 290 | 260 | 230 | 200 | 150 | 30 | 140 | |
| | 500 | 510 | 510 | 490 | 410 | 370 | 330 | 310 | 350 | 280 | 150 | 260 |
| | 480 | 500 | 500 | 510 | 500 | 460 | 430 | 260 | 150 | 230 | 280 | 360 |
| | 370 | 390 | 400 | 430 | 450 | 450 | 400 | 210 | 10 | 110 | 280 | 230 |
| | 260 | 260 | 250 | 260 | 330 | 330 | 310 | 130 | | | 130 | 130 |
| | 200 | 200 | 150 | 160 | 170 | 220 | 230 | 130 | | | | |
| | 130 | 130 | 90 | 80 | 90 | 110 | 140 | 30 | | | | |
| | 10 | 20 | 30 | 10 | 10 | 20 | 30 | | | | | |
16. Depressions below p in the S.E. quarter.
| Rings | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| | | | | | | | | | | | | |
| 7 | | | | | | | | | | | | |
| 8 | 20 | 30 | 30 | | | | | | | | | |
| 9 | 260 | 290 | 290 | 280 | 240 | 150 | 30 | | | | | |
| 10 | 420 | 440 | 450 | 440 | 420 | 370 | 270 | 140 | | | | |
| 11 | 530 | 540 | 560 | 560 | 550 | 480 | 430 | 330 | 150 | 40 | 40 | 430 |
| 12 | 500 | 510 | 520 | 550 | 630 | 600 | 500 | 430 | 290 | 230 | 200 | 630 |
| 13 | 450 | 430 | 420 | 410 | 430 | 570 | 630 | 530 | 430 | 480 | 340 | 710 |
| 14 | 360 | 330 | 310 | 290 | 280 | 330 | 510 | 670 | 590 | 630 | 50 | 830 |
| 15 | 240 | 230 | 220 | 200 | 180 | 200 | 330 | 530 | 770 | 760 | 710 | 870 |
| 16 | 180 | 160 | 150 | 130 | 110 | 140 | 230 | 330 | 630 | 830 | 790 | 880 |
| 17 | 110 | 80 | 50 | 40 | 30 | 90 | 190 | 280 | 500 | 860 | 830 | 860 |
| 18 | 10 | | | | | | | | | | | |
| 19 | | | | | | | | | | | | |
| 20 | | | | | | | | | | | | |
It remains now to find the sines of the vertical angles subtended by all the foregoing altitudes and depressions, since the sum of these sines is what we are in quest of. Each altitude or depression is the perpendicular of a right-angled triangle, of which the given radius standing on the same line with it in the right-hand margin is the base, or other side about the right-angle; and by the resolution of the right-angled triangle, for each perpendicular, the same number of corresponding sines will be found. But with such data the tangent of the angle is much easier to be found than the sine, and the analogy for that purpose is this, as the base : to the perpendicular :: 1 (radius) : the tangent required, which will therefore
fore be found by barely dividing the given perpendicular by the base; and if we find this number in its proper column in a table of sines and tangents, on the same line with it, in the column of sines will be found the sine of the angle required. This seems to be the easiest way of resolving all the triangles when computed separately. But as the labour would be very great in performing so many hundreds of arithmetical divisions, &c. either by logarithms, or by the natural numbers, instead of it, the following method, proposed by the Hon. Mr. Cavendish, was adopted, as being a much more expeditious way of obtaining the sum of the sines required. This method consists in finding, in a very easy manner, the difference between each tangent and its corresponding sine, from the given base and perpendicular, and then, subtracting the sum of all the differences from the sum of the tangents, there remains the sum of the sines. Several advantages attend this method of proceeding: for, to find the tangents we need not divide every perpendicular separately by its corresponding base, but add together all the perpendiculars that are on the same line, and divide their sum by their common base, which is the radius of the middle of the ring, and is placed on the same line with them towards the right-hand; for thus we shall have little more than a twelfth part of the number of divisions
divisions to perform: also a great part of the tangents are so small that they do not at all differ from their corresponding sines in the number of decimals that it is necessary to continue the computations to, in all which cases the trouble of finding the differences is saved; and those differences which it is necessary to compute, are very readily found by inspection on a peculiar kind of sliding rule, which was constructed for this purpose, and of which I shall here give a short description.
This rule (the figure of which is represented tab. x. fig. 2.) consists of three columns; one marked AF or base, which is moveable by sliding it up or down by the side of the other two which are fixed; of these two the one contains the perpendicular altitudes or depressions, and the other the differences between the sines and tangents to the radius r. To construct the numbers on this rule; form a series of logarithmic tangents in arithmetical progression, of which the first term is $9 \cdot 000$, and the common difference $0 \cdot 25$; take out from a table the corresponding natural tangents, and place them in the first and second columns of base and perpendicular, and the difference between the natural sine and natural tangent in the last column, marked Diff. To make use of this scale; look out any base and its corresponding perpendicular in their proper columns, that is, any radius and its cor-
corresponding altitude or depression in the fifteen foregoing tables, without regarding the number of places they contain, and bring them to correspond; then, if they consist of the same number of places, the lower index on the slider or first column, or that answering to 1000, points to the true difference between the sine and tangent in the last column; but if the number of places in the base exceed that in the perpendicular by one, the upper index 100 must be used. And in this manner were computed all the differences which were necessary to be found, and placed in their proper squares formed by the meeting of the horizontal and vertical lines, or rings and sectoral spaces, in the following set of sixteen tables, which correspond to the foregoing set of sixteen, each to each, according to the number of them, and marked at the tops with the numbers 1, 2, 3 &c. to 12 for the sectoral spaces, and with the number of the rings on the left-hand margin. Also, in the column immediately after the number of the ring are placed the radii which formed the last column in the preceding tables; then, in the third column, are placed the sums of the altitudes and depressions found in each line of the former tables; and, in the next column, the quotients found by dividing the numbers in the third by those in the second column; these quotients are the sums of the tangents belonging
belonging to each line or ring, which being all added together, their total is placed at the bottom of the column; after this follow the twelve columns of differences before mentioned, which are succeeded by one more column containing the sums of each line of these differences, which sums being added together, their total is placed at the bottom of them; and this total is the sum of all the differences between the sines and the tangents, and it is therefore subtracted from the total of the tangents in the fourth column, when there remains the sum of the sines as required.
| Rings | Radii | Sum of Perpen. | 3 ÷ 2 = Sum of Tang. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum of Diff. |
|-------|-------|----------------|---------------------|---|---|---|---|---|---|---|---|---|----|----|----|---------------|
| 1 | 3333 | 2175 | 6°525 | 103| 103| 103| 103| 97 | 97 | 86 | 75 | 55 | 36 | 20 | 5 | 883 |
| 2 | 1000 | 6265 | 6°265 | 88 | 90 | 88 | 85 | 84 | 82 | 78 | 75 | 56 | 40 | 20 | 4 | 790 |
| 3 | 1667 | 10045 | 6°027 | 79 | 86 | 87 | 87 | 88 | 97 | 93 | 64 | 48 | 21 | 7 | 1 | 758 |
| 4 | 2333 | 9300 | 3°986 | 12 | 12 | 13 | 17 | 23 | 27 | 39 | 43 | 45 | 16 | 4 | 251 |
| 5 | 3000 | 7275 | 2°425 | 1 | 1 | 2 | 3 | 6 | 11 | 16 | 27 | 13 | 3 | | 83 |
| 6 | 3667 | 4695 | 1°280 | | | | | | | | | | | | | 23 |
| 7 | 4333 | 2735 | 0°631 | | | | | | | | | | | | | 6 |
| 8 | 5000 | 1625 | 0°325 | | | | | | | | | | | | | 2 |
| 9 | 5667 | 670 | 0°118 | | | | | | | | | | | | | |
| 10 | 6333 | 60 | 0°009 | | | | | | | | | | | | | |
\[27°59' = \text{sum of tangents.}\]
\[2°796 = \text{sum of the diff.}\]
\[24°795 = \text{sum of the sines of alt. above o in the N.W. quarter.}\]
2. For the sum of the fines above 0 in the N.E. quarter.
| Rings | Radii | Sum of Perp. | 3 ÷ 2 = Sum of Tang. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum of Diff. |
|-------|-------|--------------|----------------------|---|---|---|---|---|---|---|---|---|---|----|----|-------------|
| 1 | 333½ | 1965 | 5·895 | 97| 92| 92| 87| 80| 71| 56| 43| 33| 24| 15| 5 | 6·95 |
| 2 | 1000 | 5360 | 5·360 | 68| 67| 65| 62| 59| 56| 53| 43| 35| 23| 9 | 1 | 541 |
| 3 | 1667 | 8275 | 4·965 | 66| 53| 51| 49| 47| 41| 36| 31| 25| 20| 13| 1 | 433 |
| 4 | 2333 | 7555 | 3·238 | 10| 10| 10| 10| 11| 12| 14| 15| 13| 8 | 1 | 124 |
| 5 | 3000 | 4410 | 1·470 | | | | | | | | | | | | 13 |
| 6 | 3667 | 1937 | 0·528 | | | | | | | | | | | | 2 |
| 7 | 4333 | 580 | 0·134 | | | | | | | | | | | | |
| 8 | 5000 | 50 | 0·010 | | | | | | | | | | | | |
21·600 = sum of the tangents.
1·808 = sum of the differences.
19·792 = sum of the fines.
3. For the sum of the fines above 0 in the S.W. quarter.
| Rings | Radii | Sum of Perp. | 10 | 0·30 |
|-------|-------|--------------|----|-----|
| 1 | 333½ | 10 | 0·30 |
| 2 | 1000 | 15 | 15 |
| 3 | 1667 | 5 | 3 |
| 12 | 7667 | 40 | 5 |
| 13 | 8333 | 60 | 7 |
| 14 | 9000 | 100 | 11 |
| 15 | 9667 | 200 | 21 |
| 16 | 10333 | 1960 | 190 |
| 17 | 11000 | 3720 | 338 |
| 18 | 11667 | 5770 | 495 |
| 19 | 12333 | 7260 | 573 |
| 20 | 13000 | 8920 | 686 |
2·374 = sum of the tangents, or sum of the fines, as the diff. between them are nothing in this quadrant.
4. For the sum of the fines above 0 in the S.E. quarter.
| Rings | Radii | Sum of Perp. | 10 | 0·001 |
|-------|-------|--------------|----|------|
| 16 | 10333 | 10 | 0·001 |
| 17 | 11000 | 230 | 21 |
| 18 | 11667 | 830 | 71 |
| 19 | 12333 | 1450 | 118 |
| 20 | 13000 | 2130 | 164 |
0·375 = sum of the tangents, or of the fines, the diff. being nothing.
### 5. For the fines below 0 in the N.W. quarter.
| Rings | Radii | Sum of Perp. | $\frac{3}{2} = \text{Sum of Tang.}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum of Diff. |
|-------|-------|--------------|-----------------------------------|---|---|---|---|---|---|---|---|---|---|----|----|-----|
| 6 | 3667 | 145 | 0'040 | | | | | | | | | | | | |
| 7 | 4333 | 970 | 224 | | | | | | | | | | | | |
| 8 | 5000 | 2370 | 474 | | | | | | | | | | | | |
| 9 | 5667 | 4325 | 763 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 10 | 6333 | 5910 | 933 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 11 | 7000 | 7520 | 1'074 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 12 | 7667 | 9280 | 1'210 | 2 | 1 | 1 | 1 | 1 | | | | | | | |
| 13 | 8333 | 10140 | 1'219 | 2 | 1 | 1 | 1 | 1 | | | | | | | |
| 14 | 9000 | 10700 | 1'078 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 15 | 9667 | 11580 | 1'198 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 16 | 10333 | 11970 | 1'159 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 17 | 11000 | 12250 | 1'105 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 18 | 11667 | 12280 | 1'052 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 19 | 12333 | 12750 | 1'034 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 20 | 13000 | 13940 | 1'072 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
$13'635 =$ sum of the tangents.
$'101 =$ sum of the differences.
$13'534 =$ sum of the fines.
### 6. For the sum of the fines below 0 in the N.E. quarter.
| | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-------|-------|-------|---|---|---|---|---|---|---|---|---|----|----|-----|
| 6 | 3667 | 180 | 0'049 | | | | | | | | | | | | |
| 7 | 4333 | 1160 | 268 | | | | | | | | | | | | |
| 8 | 5000 | 2980 | 596 | | | | | | | | | | | | |
| 9 | 5667 | 5210 | 919 | 1 | 1 | 1 | 1 | | | | | | | |
| 10 | 6333 | 7300 | 1'153 | 1 | 1 | 1 | 1 | | | | | | | |
| 11 | 7000 | 9440 | 1'349 | 1 | 2 | 2 | 2 | 1 | | | | | | | |
| 12 | 7667 | 11460 | 1'495 | 2 | 2 | 2 | 2 | 1 | | | | | | | |
| 13 | 8333 | 13080 | 1'570 | 1 | 1 | 1 | 2 | 2 | 1 | | | | | | |
| 14 | 9000 | 13290 | 1'477 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 15 | 9667 | 12670 | 1'311 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 16 | 10333 | 12160 | 1'177 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
| 17 | 11000 | 11850 | 1'077 | 1 | 1 | 1 | 1 | 1 | | | | | | | |
$12'441 =$ sum of the tangents.
$'085 =$ sum of the differences.
$12'356 =$ sum of the fines.
7. For the sum of the fines below 0 in the S.W. quarter.
| Rings | Radii. | Sum of Dep. | $3 \div 2 =$ Sum of Tang. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum of Diff. |
|-------|--------|-------------|--------------------------|---|---|---|---|---|---|---|---|---|---|----|----|-----|
| 1 | 333$\frac{1}{2}$ | 1445 | 4'335 | 51 | 51 | 47 | 43 | 40 | 35 | 27 | 21 | 17 | 10 | 2 | '344 |
| 2 | 1000 | 3460 | 3'460 | 28 | 27 | 26 | 25 | 19 | 17 | 13 | 9 | 6 | 5 | 2 | 177 |
| 3 | 1667 | 5370 | 3'222 | 21 | 19 | 17 | 15 | 13 | 18 | 16 | 6 | 5 | 4 | 2 | 136 |
| 4 | 2333 | 6650 | 2'850 | 15 | 14 | 13 | 12 | 10 | 8 | 8 | 5 | 5 | 3 | 2 | 95 |
| 5 | 3000 | 7640 | 2'547 | 9 | 9 | 9 | 8 | 6 | 6 | 5 | 4 | 3 | 3 | 2 | 64 |
| 6 | 3667 | 8260 | 2'253 | 5 | 5 | 5 | 5 | 5 | 5 | 3 | 3 | 3 | 2 | 1 | 44 |
| 7 | 4333 | 8260 | 1'906 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 2 | 2 | 1 | 25 |
| 8 | 5000 | 7840 | 1'568 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 15 |
| 9 | 5667 | 6580 | 1'161 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 11 |
| 10 | 6333 | 5680 | 897 | | | | | | | | | | | | | 5 |
| 11 | 7000 | 4450 | 636 | | | | | | | | | | | | | 3 |
| 12 | 7667 | 3160 | 412 | | | | | | | | | | | | | 2 |
| 13 | 8333 | 2190 | 263 | | | | | | | | | | | | | 1 |
| 14 | 9000 | 1250 | 139 | | | | | | | | | | | | | |
| 15 | 9667 | 570 | 59 | | | | | | | | | | | | | |
| 16 | 10333 | 210 | 20 | | | | | | | | | | | | | |
$25'728 = \text{sum of the tangents.}$
$0'922 = \text{sum of the differences.}$
$24'806 = \text{sum of the fines.}$
8. For the sum of the fines below 0 in the S.E. quarter.
| Rings | Radii | Sum of Dep. | $3 \div 2 =$ Sum of Tang. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum of Diff. |
|-------|-------|-------------|--------------------------|---|---|---|---|---|---|---|---|---|---|----|----|--------|
| 1 | 333$\frac{1}{2}$ | 1510 | 4'530 | 51 | 51 | 47 | 43 | 40 | 35 | 27 | 21 | 17 | 13 | 11 | 1 | '357 |
| 2 | 1000 | 4120 | 4'120 | 28 | 28 | 28 | 28 | 28 | 27 | 25 | 21 | 17 | 13 | 8 | 1 | 252 |
| 3 | 1667 | 6510 | 3'906 | 21 | 21 | 21 | 21 | 21 | 21 | 19 | 16 | 13 | 9 | 1 | 205 |
| 4 | 2333 | 8070 | 3'459 | 16 | 16 | 15 | 15 | 15 | 15 | 14 | 14 | 13 | 10 | 6 | 1 | 150 |
| 5 | 3000 | 8990 | 2'997 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 8 | 5 | 1 | 95 |
| 6 | 3667 | 9280 | 2'531 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 6 | 4 | 1 | 57 |
| 7 | 4333 | 8440 | 1'948 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 3 | 1 | 27 |
| 8 | 5000 | 7490 | 1'498 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 14 |
| 9 | 5667 | 6630 | 1'170 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 |
| 10 | 6333 | 6070 | 958 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 |
| 11 | 7000 | 5110 | 730 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 |
| 12 | 7667 | 4210 | 549 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| 13 | 8333 | 3540 | 425 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| 14 | 9000 | 3370 | 374 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| 15 | 9667 | 3020 | 313 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| 16 | 10333 | 2680 | 259 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| 17 | 11000 | 2310 | 210 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 18 | 11667 | 1975 | 169 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 19 | 12333 | 1730 | 140 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 20 | 13000 | 1520 | 117 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$30'403 =$ sum of the tangents.
$1'190 =$ sum of the differences.
$29'213 =$ sum of the fines.
9. For the sum of the fines above p in the N.W. quarter.
| Rings | Radii | Sum of Dep. | $3 \div 2 =$ Sum of Tang. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum of Diff. |
|-------|-------|-------------|--------------------------|---|---|---|---|---|---|---|---|---|---|----|----|--------|
| 4 | 2333 | 10 | '004 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 5 | 3000 | 15 | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 6 | 3667 | 15 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 7 | 4333 | 15 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 8 | 5000 | 15 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$0'019 =$ sum of the tangents, or sum of the fines, as they have no difference in this quadrant.
10. For the sum of the fines above p in the N.E. quarter.
| Rings | Radii | Sum of Dep. | $3 \div 2 =$ Sum of Tang. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum of Diff. |
|-------|-------|-------------|--------------------------|---|---|---|---|---|---|---|---|---|---|----|----|--------|
| 1 | 333$\frac{1}{2}$ | 10 | '030 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1000 | 10 | 10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 3 | 1667 | 15 | 9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 4 | 2333 | 60 | 26 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 5 | 3000 | 40 | 13 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 6 | 3667 | 5 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$0'090 =$ sum of the tangents, or sum of the fines, they being equal in this quadrant.
11. For the sum of the fines above p in the S.W. quarter.
| Rings | Radii | Sum of Alt. | 3½ = Sum of Tang. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum of Diff. |
|-------|-------|-------------|-------------------|---|---|---|---|---|---|---|---|---|----|----|----|-------------|
| 1 | 33½ | 1110 | 3½30 | 17| 17| 15| 15| 13| 13| 12| 12| 10| 8 | 6 | 1 | *139 |
| 2 | 1000 | 3150 | 3½150 | 18| 17| 16| 14| 13| 12| 11| 10| 6 | 5 | 2 | 1 | 125 |
| 3 | 1667 | 6780 | 4½068 | 28| 28| 28| 28| 28| 27| 23| 20| 18| 14| 5 | 1 | 248 |
| 4 | 2333 | 10930 | 4½684 | 37| 38| 39| 40| 41| 42| 38| 34| 26| 18| 7 | 1 | 361 |
| 5 | 3000 | 13740 | 4½580 | 18| 23| 34| 33| 38| 39| 37| 34| 33| 26| 14| 2 | 331 |
| 6 | 3667 | 11240 | 3½065 | 3 | 5 | 5 | 6 | 6 | 7 | 14| 16| 19| 21| 14| 2 | 118 |
| 7 | 4333 | 7230 | 1½668 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 5 | 2 | 20 |
| 8 | 5000 | 3900 | 780 | | | | | | | | | | | | | 5 |
| 9 | 5667 | 1560 | 275 | | | | | | | | | | | | | 1 |
| 10 | 6333 | 440 | 70 | | | | | | | | | | | | | 1 |
| 15 | 9667 | 310 | 32 | | | | | | | | | | | | | 1 |
| 16 | 10333 | 640 | 62 | | | | | | | | | | | | | 1 |
| 17 | 11000 | 970 | 88 | | | | | | | | | | | | | 1 |
| 18 | 11667 | 1590 | 137 | | | | | | | | | | | | | 1 |
| 19 | 12333 | 2480 | 201 | | | | | | | | | | | | | 1 |
| 20 | 13000 | 3130 | 241 | | | | | | | | | | | | | 2 |
26·431 = sum of the tangents.
1·353 = sum of the differences.
25·078 = sum of the fines.
12. For the sum of the fines above p in the S.E. quarter.
| Rings | Radii | Sum of Alt. | 3½ = Sum of Tang. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-------|-------------|-------------------|---|---|---|---|---|---|---|---|---|----|----|----|
| 1 | 33½ | 1110 | 3½30 | 17| 17| 15| 15| 13| 13| 12| 12| 10| 8 | 6 | 1 |
| 2 | 1000 | 3150 | 3½150 | 18| 17| 16| 14| 13| 12| 11| 10| 6 | 5 | 2 |
| 3 | 1667 | 5890 | 3½534 | 28| 26| 23| 21| 18| 16| 14| 12| 9 | 5 | 2 |
| 4 | 2333 | 8420 | 3½609 | 34| 33| 30| 26| 24| 18| 15| 12| 6 | 4 | 1 |
| 5 | 3000 | 9440 | 3½147 | 18| 18| 18| 18| 18| 18| 16| 9 | 3 | 2 | 1 |
| 6 | 3667 | 8260 | 2½253 | 3 | 4 | 5 | 5 | 6 | 6 | 8 | 7 | 4 | 3 | 1 |
| 7 | 4333 | 5530 | 1½276 | 1 | 1 | 1 | 2 | 3 | 3 | 1 | | | | |
| 8 | 5000 | 2280 | 456 | | | | | | | | | | | | |
| 9 | 5667 | 1060 | 187 | | | | | | | | | | | | |
| 10 | 6333 | 370 | 58 | | | | | | | | | | | | |
| 18 | 11667 | 150 | 13 | | | | | | | | | | | | |
| 19 | 12333 | 510 | 42 | | | | | | | | | | | | |
| 20 | 13000 | 700 | 54 | | | | | | | | | | | | |
21·109 = sum of the tangents.
0·848 = sum of the differences.
20·261 = sum of the fines.
### 13. For the sum of the fines below p in the N.W. quarter.
| Rings | Radii. | Sum of Dep. | 3-2 = Sum of Tang. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum of Diff |
|-------|--------|-------------|-------------------|---|---|---|---|---|---|---|---|---|----|----|----|------------|
| 1 | 333½ | 790 | 2°370 | 13 | 12 | 9 | 8 | 7 | 5 | 5 | 3 | 2 | 1 | 1 | 0°66 |
| 2 | 1000 | 3170 | 3°170 | 26 | 25 | 21 | 17 | 14 | 12 | 9 | 6 | 5 | 3 | 2 | 14° |
| 3 | 1667 | 4980 | 2°988 | 15 | 14 | 13 | 12 | 11 | 9 | 8 | 7 | 5 | 2 | 108 |
| 4 | 2333 | 6200 | 2°657 | 10 | 10 | 9 | 9 | 8 | 7 | 6 | 5 | 5 | 4 | 2 | 75 |
| 5 | 3000 | 7270 | 2°423 | 10 | 10 | 8 | 7 | 6 | 5 | 5 | 4 | 3 | 2 | 1 | 61 |
| 6 | 3667 | 7800 | 2°127 | 7 | 6 | 5 | 5 | 4 | 3 | 3 | 3 | 2 | 1 | 1 | 40 |
| 7 | 4333 | 7975 | 1°840 | 5 | 4 | 4 | 3 | 3 | 2 | 2 | 1 | 1 | 1 | 1 | 27 |
| 8 | 5000 | 8280 | 1°656 | 3 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 17 |
| 9 | 5667 | 8660 | 1°528 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 14 |
| 10 | 6333 | 8935 | 1°411 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 11 |
| 11 | 7000 | 9580 | 1°369 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 9 |
| 12 | 7667 | 10280 | 1°341 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 9 |
| 13 | 8333 | 11200 | 1°344 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 |
2°224 = sum of the tangents.
0°587 = sum of the differences.
25°637 = sum of the fines.
### 14. For the sum of the fines below p in the N.E. quarter.
| Rings | Radii. | Sum of Dep. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | '032 |
|-------|--------|-------------|---|---|---|---|---|---|---|---|---|----|----|----|-----|
| 1 | 333½ | 625 | 1°875 | 7 | 5 | 5 | 4 | 3 | 2 | 2 | 1 | 1 | 1 | 1 | 98 |
| 2 | 1000 | 2755 | 2°755 | 17 | 16 | 15 | 13 | 11 | 9 | 6 | 5 | 3 | 2 | 1 | 104 |
| 3 | 1667 | 4855 | 2°913 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 6 | 4 | 2 | 98 |
| 4 | 2333 | 7000 | 3°000 | 11 | 12 | 13 | 13 | 13 | 13 | 11 | 9 | 6 | 4 | 2 | 107 |
| 5 | 3000 | 8500 | 2°833 | 11 | 11 | 11 | 11 | 11 | 10 | 10 | 9 | 7 | 5 | 2 | 98 |
| 6 | 3667 | 9580 | 2°613 | 6 | 7 | 7 | 8 | 8 | 8 | 6 | 5 | 4 | 2 | 69 |
| 7 | 4333 | 10350 | 2°389 | 5 | 5 | 5 | 4 | 4 | 4 | 5 | 5 | 5 | 4 | 2 | 49 |
| 8 | 5000 | 11840 | 2°368 | 3 | 2 | 2 | 2 | 2 | 1 | 2 | 3 | 3 | 3 | 1 | 27 |
| 9 | 5667 | 8400 | 1°482 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 14 |
| 10 | 6333 | 7610 | 1°202 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 9 |
| 11 | 7000 | 7930 | 1°133 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 7 |
| 12 | 7667 | 8810 | 1°149 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |
| 13 | 8333 | 8980 | 1°078 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 7 |
2°790 = sum of the tangents.
6°29 = sum of the differences.
26°161 = sum of the fines.
15. For the sum of the fines below p in the S.W. quarter.
| Rings | Radii | Sum of Dep. | $3 \div 2 =$ Sum of Tang. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum of Diff. |
|-------|-------|-------------|--------------------------|---|---|---|---|---|---|---|---|---|---|----|----|--------|
| 9 | 5667 | 600 | '051 | | | | | | | | | | | |
| 10 | 6333 | 1730 | 273 | | | | | | | | | | | |
| 11 | 7000 | 3030 | 433 | | | | | | | | | | | |
| 12 | 7667 | 4470 | 583 | | | | | | | | | | | |
| 13 | 8333 | 4660 | 559 | | | | | | | | | | | |
| 14 | 9000 | 3730 | 414 | | | | | | | | | | | |
| 15 | 9667 | 2390 | 247 | | | | | | | | | | | |
| 16 | 10333 | 1460 | 141 | | | | | | | | | | | |
| 17 | 11000 | 800 | 73 | | | | | | | | | | | |
| 18 | 11667 | 130 | 11 | | | | | | | | | | | |
$2'785 =$ sum of the tangents.
$'011 =$ sum of the differences.
$2'774 =$ sum of the fines.
16. For the sum of the fines below p in the S.E. quarter.
| | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|-------|-----------|---|---|---|---|---|---|---|---|---|----|----|----|
| 7 | 4333 | 80 | 018 | | | | | | | | | | |
| 8 | 5000 | 290 | 58 | | | | | | | | | | |
| 9 | 5667 | 1810 | 319 | | | | | | | | | | |
| 10 | 6333 | 3280 | 518 | | | | | | | | | | |
| 11 | 7000 | 4640 | 663 | | | | | | | | | | |
| 12 | 7667 | 5590 | 729 | | | | | | | | | | |
| 13 | 8333 | 5830 | 700 | | | | | | | | | | |
| 14 | 9000 | 5700 | 633 | | | | | | | | | | |
| 15 | 9667 | 5240 | 542 | | | | | | | | | | |
| 16 | 10333 | 4560 | 441 | | | | | | | | | | |
| 17 | 11000 | 3920 | 356 | | | | | | | | | | |
| 18 | 11667 | 3180 | 273 | | | | | | | | | | |
| 19 | 12333 | 2630 | 213 | | | | | | | | | | |
| 20 | 13000 | 2230 | 172 | | | | | | | | | | |
$5'635 =$ sum of the tangents.
$'025 =$ sum of the differences.
$5'610 =$ sum of the fines.
Having now obtained the sums of the fines for the several quadrants, the next business is to collect them together, and deduct the negatives from the affirmatives. And this may be done either for each observatory separately, or for both together. I shall do them separately, in order thereby to discover also the ratio of their effects.
And, first, for the Southern observatory o.
Affirmatives.
1.. 24°795 N.W.
2.. 19°792 N.E.
7.. 24°806 S.W.
8.. 29°213 S.E.
Negatives.
3.. 2°374 S.W.
4.. 0°375 S.E.
5.. 13°534 N.W.
6.. 12°356 N.E.
98°606 = sum of affirm.
28°639 = sum of negat.
69°967 = effective sum of the fines for o.
Secondly, for the Northern observatory p.
Affirmatives.
11.. 25°078 S.W.
12.. 20°261 S.E.
13.. 25°637 N.W.
14.. 26°161 N.E.
Negatives.
9.. 0°019 N.W.
10.. 0°090 N.E.
15.. 2°774 S.W.
16.. 5°610 S.E.
97°137 = sum of affirm.
8°493 = sum of negat.
88°644 = effective sum of the fines for p.
69°967 = the same for o.
158°611 = the sum of the fines for both observ.
From these numbers it appears, that the effect of the attraction at the Northern observatory is to that at the Southern one, nearly as 70 is to 89, or as 7 to 9 nearly. This difference is to be attributed chiefly to the effect of the hills on the South of the Southern observatory, which were considerably greater and nearer to it than those on the back of the Northern observatory. For although the Southern observatory was placed 273 feet above the level of the Northern one, which removed it considerably more above the center of gravity of the hill than the latter was, it was at the same time placed considerably nearer than the other to the middle in a horizontal direction; so that probably the one difference nearly balances the other; and accordingly we find that the sum of the affirmative altitudes for o is $44.587$, and of those for p $45.339$, which differ by only a 45th part nearly.
It only remains now to multiply the sum of the sines by the common breadth of the rings, and by the common difference of the sines of the angles made by the meridian and the several radii. It has already been observed, that the former is $666\frac{2}{3}$, and the latter $\frac{1}{12}$; therefore $\frac{1}{12} \times 666\frac{2}{3} = \frac{2000}{36} = \frac{500}{9}$ is their product: consequently, $158.611 \times \frac{500}{9} = 8811\frac{2}{3}$ nearly, is the sum of the two opposite attractions made by the hill, &c. at the two observatories.
In order now to compare this attraction with that of the whole earth, this body may be considered as a sphere, and the observatories as placed at its surface; since the very final differences of these suppositions from the truth, are of no consequence at all in this comparison. Now the attraction of a sphere, on a body at its surface, is known to be $\frac{2}{3} cd$, where $d$ is = the diameter of the sphere, and $c = 3 \cdot 1416 =$ the circumference of the circle of which the diameter is 1. But $cd$ is = the circumference of the circle to the diameter $d$; and therefore the attraction of a sphere will be expressed by barely $\frac{2}{3}$ of its circumference; which is a theorem well adapted to the computation in hand. The length of a degree in the mean latitude of $45^\circ$, is 57028 French toises (see p. 327. Phil. Trans. 1768): and the same result nearly is obtained by taking a mean among all the measures of degrees there put down, that mean being 57038 toises. I shall therefore use the round number 57030 as probably nearer the truth. This number being multiplied by 6, the product 342180 shews the number of French feet in one degree; but, by p. 326. of the same volume, the lengths of the Paris and London feet are as 76.734 to 72, that is, as 4.263 to 4; therefore, as $4 : 4.263 :: 342180 : 364678 =$ the English feet in one degree; and this being multiplied by 360 the whole number of degrees, there results
Mr. Hutton's Calculations to ascertain
131284080 feet for the whole circumference, which are equal to 24864\(\frac{1}{2}\) miles, making \(69\frac{1}{15}\) to a degree in the mean latitude. Lastly, \(\frac{2}{3}\) of 131284080 give 87522720 for the measure of the attraction of the whole earth.
Consequently, the whole attraction of the earth is to the sum of the two contrary attractions of the hill, as the number 87522720 to 8811\(\frac{2}{3}\), that is, as 9933 to 1 very nearly, on supposition that the density of the matter in the hill is equal to the mean density of that in the whole earth.
But the Astronomer Royal found, by his observations, that the sum of the deviations of the plumb line, produced by the two contrary attractions, was 11.6 seconds. From hence it is to be inferred, that the attraction of the earth is actually to the sum of the attractions of the hill, nearly as radius to the tangent of 11.6 seconds, that is, as 1 to .000056239, or as 17781 to 1; or as 17804 to 1 nearly, after allowing for the centrifugal force arising from the rotation of the earth about its axis.
Having now obtained the two results, namely; that which arises from the actual observations, and that belonging to the computation on the supposition of an equal density in the two bodies, the two proportions compared must give the ratio of their densities, which is
that of 17804 to 9933, or 1434 to 800 nearly, or almost as 9 to 5. And so much does the mean density of the earth exceed that of the hill.
Thus then we have at length obtained the object which we have been in quest of through the very laborious calculations that have been described in this paper, and in the survey and measurements from which these computations were made; namely, the ratio of the mean density of all the matter in the earth, in comparison with the density of the matter of which the hill is composed. And that ratio we have found to be equal to the ratio of 9 to 5. And, for the reasons before mentioned, I think we may rest satisfied, that this proportion is obtained to a considerable degree of proximity, probably to within the fiftieth part, if not the hundredth part of its true magnitude. Another question, however, still arises, namely, what is the density of the matter in the hill? Is its mean density equal to that of water, of sand, of clay, of chalk, of stone, or of some of the metals? For, according to the matter, or different sorts of matter, of which it is formed, and according as it is constituted with or without large vacuities, its mean density may be greater or less, and that in a degree which is not certainly known. A considerable degree of accuracy in this point could only be obtained by a close examination of the internal structure
Mr. Hutton's Calculations to ascertain structure of the hill. And the easiest method of doing this would be to procure holes to be bored, in several parts of it, from the surface to a sufficient depth, after the manner that is practiced in boring holes to the coal mines from the surface of the ground; for by such operation it is known what kind of strata the borer is passed through, together with their dimensions and densities. The proper mean among all these would be the mean density of the hill, as compared to water or to any other simple matter; and thence we should obtain the comparative density of the whole earth with respect to water: but in the present instance, we must be satisfied with the estimate arising from the report of the external view of the hill; which is, that to all appearance it consists of an entire mass of solid rock. It is probable, therefore, that we shall not greatly err, if we assume the density of the hill equal to that of common stone; which is not much different from the mean density of the whole matter near the surface of the earth, to such depths as have actually been explored either by digging or boring. Now the density of common stone is to that of rain water as $2\frac{1}{2}$ to 1; which being compounded with the proportion of 9 to 5 above found, there results the ratio of $4\frac{1}{2}$ to 1 for the ratio of the densities of the earth and rain water;
that is to say, the mean density of the whole earth is about $4\frac{1}{2}$ times the density of water.
To what useful purposes the knowledge of the mean density of the earth, as above found, may be applied, it is not my business here to shew. I shall therefore put an end to this paper with a reflection or two on the premises before delivered. Sir Isaac Newton thought it probable, that the mean density of the earth might be five or six times as great as the density of water; and we have now found, by experiment, that it is very little less than what he had thought it to be: so much justness was even in the premises of this wonderful man! Since then the mean density of the whole earth is about double that of the general matter near the surface, and within our reach, it follows, that there must be somewhere within the earth, towards the more central parts, great quantities of metals, or such like dense matter, to counterbalance the lighter materials, and produce such a considerable mean density. If we suppose, for instance, the density of metal to be 10, which is about a mean among the various kinds of it, the density of water being 1, it would require sixteen parts out of twenty-seven, or a little more than one-half of the matter in the whole earth, to be metal of this density, in order to compose a mass of such mean density as we have found the earth to possess by
Mr. Hutton's Calculations to ascertain our experiment: or $\frac{4}{15}$, or between $\frac{1}{3}$ and $\frac{1}{4}$ of the whole magnitude will be metal; and consequently $\frac{2}{3}$, or nearly $\frac{2}{3}$ of the diameter of the earth, is the central or metalline part.
Knowing then the mean density of the earth in comparison with water, and the densities of all the planets relatively to the earth, we can now assign the proportions of the densities of all of them as compared to water, after the manner of a common table of specific gravities. And the numbers expressing their relative densities, in respect of water, will be as below, supposing the densities of the planets, as compared to each other, to be as laid down in Mr. De La Lande's astronomy.
| Planet | Density |
|--------------|---------|
| Water | 1 |
| The Sun | $1\frac{2}{15}$ |
| Mercury | $9\frac{1}{6}$ |
| Venus | $5\frac{1}{5}$ |
| The earth | $4\frac{1}{2}$ |
| Mars | $3\frac{2}{7}$ |
| The Moon | $3\frac{1}{15}$ |
| Jupiter | $1\frac{1}{24}$ |
| Saturn | $\frac{13}{32}$ |
Thus then we have brought to a conclusion the computation of this important experiment, and, it is hoped, with no inconsiderable degree of accuracy. But it is the first
first experiment of the kind which has been so minutely and circumstantially treated; and first attempts are seldom so perfect and just as succeeding endeavours afterwards render them. And, besides, a frequent repetition of the same experiment, and a coincidence of results, afford that firm dependance on the conclusions and satisfaction to the mind, which can scarcely ever be had from a single trial, however carefully it may be executed. For those reasons it is to be wished, that the world may not rest satisfied barely with what has been done in this instance, but that they will repeat the experiment in other situations, and in other countries, with all the care and precision that it may be possible to give to it, till an uniformity of conclusions shall be found, sufficient to establish the point in question beyond any reasonable possibility of doubt. What has been already done in the present case will render any future repetition more easy and perfect. But improvements may be made, perhaps both in the mode of computation and in the survey; in the latter, especially, there certainly may. Some improvements of this kind I have hinted at in some parts of this paper, which with others I shall here collect together, that they may readily be seen in one point of view. They are principally these. Procure one base, or more if convenient, very accurately measured, in such situation, that
as many more points as possible in the survey may be seen from it. Assume as many principal or eminent points and objects as may be proper and convenient; and from each one of them measure the angles formed by all the rest that can be seen, both horizontal and vertical angles, and repeat these observations, if convenient, with the instrument varied or reversed, taking the means among the several quantities of each angle. Take then as many sections of the ground, and as far extended in all directions, as the time and circumstances will possibly admit. Of the sections, those that are horizontal or level are the best, as they require no calculation; procure therefore as many as possible of them. In vertical sections observe the vertical angles, not in the plane of the section, but at some other point of which the bearing is also taken from the beginning of the section line, and where the horizontal angles of the poles are taken, for the reasons before mentioned in p. 723. And it will be a still farther convenience if the section be made in such direction as to form a right angle with the line drawn to the point or station from whence the vertical angles of the poles are observed, as may be seen from what is said in p. 721. It might, perhaps, be proper to make some experiments on a valley instead of a hill, taking two observatories at the two opposite sides of it, both for the greater
greater variety in this interesting problem, and because also the survey would be more easily made, on account of the ground being more in view at each station than in the case of a hill, which generally hides more than half the compass from the observer. In computing the relative altitudes of all the principal stations, let the operations be performed mutually both backwards and forwards, that is, from both of every two objects, having for that purpose observed at each of them the vertical angle of the other, namely, both the angle of elevation and the angle of depression, and take the mean between the two computed differences of altitude; for this excludes the necessity of making the proper allowances for refraction, and for the curvature of the earth; since the effect of each of these is balanced and corrected by that of the counter observation. But as to those points in the sections which are far distant from the observer, and where great accuracy is required, it may be proper to make the allowance for refraction and curvature, as there is generally no back observation by which their effects may be balanced. These are the chief hints which at present occur to me, besides the general information to be derived by the computer from the perusal of the modes of computation that have been described in this paper. As to the surveyor, he will strike out other convenient
convenient ways of measurement adapted to the circumstances with which the nature of the survey may happen to be attended.
A map of the country about Schehallien is hereunto annexed, to convey a general idea of the nature of the ground, and for the better illustration of the description given in the former parts of this paper. This map is tab. xi.
Woolwich,
April 27, 1778.
A Sketch of Schehallien,
With Part of the Hills,
and other Places adjacent.