The Construction of the Logarithmic Lines on the Gunter's Scale; By Mr. John Robertson, F. R. S.

sides; and the clear water only is from the surface of the basin let out into the pits. If there be any thing, wherein I may further satisfy your lordship's inquiries in this or any other matter, your commands shall most cheerfully be obey'd by Your lordship's much obliged, and most obedient humble servant, William Henry. XIII. The Construction of the logarithmic Lines on the Gunter's Scale; by Mr. John Robertson, F. R. S. Read June 18, 1752. HAVING lately had occasion to treat on the construction of the Gunter's scale, I searched several books, wherein I suspected were contained the reasons of the common methods of laying down the logarithmic lines usually put on those scales: but not finding, either from my own search, or that of my friends, any satisfactory account of this matter, I drew up the following paper, to be laid before the Royal Society. The Gunter's scale * is an instrument almost universally known, and amply described by many writers; therefore * So called from its inventor Mr. Edmund Gunter, astronomy-professor in Gresham-College, from March 6, 1619, till his death, Dec. 10, 1626. therefore I shall not take up your time in useless repetitions, but only shew, on what principles the divisions of the logarithmic sines, tangents, and versed sines, are usually protracted. The line of numbers on these scales consists of two equal lengths, commonly called two radii; the first containing the logarithms of numbers from 10 to 100; and in the second are inserted those between 100 and 1000, or such of them, as can conveniently be introduced. These divisions are taken from a scale of equal parts; such, that 100 make the length of one radius; and from this scale, the divisions for the sines, tangents, and versed sines, are also taken. Now, from this construction of the line of numbers, it is plain, that, as the numbers in one radius exceed those in the other, by one place in the scale of numeration; therefore the difference of their indices must also be unity: so that such numbers only, whose index differs by 1, can be estimated in a length of two radii: but, in a length of three radii, numbers, whose indices differ by 2, may be read; and a difference of 3 may be reckon'd in a length of 4 radii, &c. The tables of logarithmic sines, tangents, secants, and versed sines, are generally computed for a circle, whose radius is 10,000,000: therefore, In the sines, the index 9 belongs to all between | Index | Range | |-------|---------------| | 8 | 5 44 36 and 0 34 23 | | 7 | 0 34 23 and 0 3 27 | &c. N In In the tangents, the index 9 belongs to all between $45^\circ$ and $5^\circ 42' 10''$. And the indices, 8, 7, 6, &c., fall as in the fines. In the versed fines, the index 10 belongs to all between $180^\circ$ and $90^\circ$, $90^\circ$ and $25^\circ 51'$, $25^\circ 51'$ and $8^\circ 7'$, $8^\circ 7'$ and $2^\circ 34'$, $2^\circ 4'$ and $0^\circ 45'$, &c. Now, as the length of the Gunter's scale admits of no more than two radii, or of such numbers only, whose index differs by unity; therefore, within this length, no more of the fines, tangents, or versed fines, can be introduced, than those, whose index differs by unity: And as not only the greatest number among the fines and tangents, but also those more generally wanted, have the indices 9 and 8 differing by unity; therefore all the fines from $90^\circ$ to $0^\circ 34'$, and all the tangents from $45^\circ$ to $0^\circ 34'$, are those only, which are put on these scales; the divisions answering to the lesser fines and tangents being omitted for want of room. And this is the reason, why the fine of $90^\circ$, and the tangent of $45^\circ$, are limited by the same termination as the second radius on the line of numbers. To construct the line of logarithmic fines. From the scale of equal parts, take the numbers expressing the arithmetical complements of the log. sines of the successive degrees, and parts of degrees, intended to be put on the scale, descending orderly from $90^\circ$: then these distances successively laid from the mark representing $90^\circ$ at the right-hand end of the scale, will give the several divisions of a scale of logarithmic sines. For, the ends of any scale being assigned, the progressive divisions of that scale are laid thereon from that end, which represents the beginning of the progression: or, the same divisions may be laid from the other end, by taking the complements of the terms to the whole length of the scale: Consequently the arithmetical complements of the sines are to be laid from the division representing $90^\circ$ degrees. To construct the line of logarithmic tangents. These are laid down in the same manner, and for the same reasons, that the sines were; the tangent of $45^\circ$ standing against the sine of $90^\circ$. The divisions for the tangents above $45^\circ$, are reckoned on the same line from $45^\circ$ towards the left-hand; or any tangent and its co-tangent are expressed by the same division. Thus one mark serves for $40^\circ$ and $50^\circ$; and the division at $30^\circ$ serves also for $60^\circ$; that at $20^\circ$ serves for $70^\circ$, &c. and the like is to be understood of the intermediate divisions. For, as the tangent of an arc, is to radius; So is radius, to the co-tangent of that arc. Therefore the tangent is equal to the square of radius divided by the co-tangent. And the co-tangent is equal to the square of radius divided by the tangent. Now the radius being unity, its square is also unity. Therefore the tangent and co-tangent of any arc are the reciprocals one of the other. But the reciprocals of numbers are correlatives to the arithmetical complements of their logarithms. Therefore the logarithms of a tangent and its co-tangent are arithmetical complements one of the other; and consequently will fall at equal distances from 45 degrees. Therefore, in the line of logarithmic tangents, the divisions to degrees under 45 serve also for those above; both being equally distant from 45 degrees. To construct the line of logarithmic versed sines. As the greatest number of degrees will fall within the limits of the scale by beginning at $180^\circ$; therefore the termination of this line is at $180^\circ$, which is put against $90^\circ$ on the sines: and although the numbers annexed to the divisions increase in the order from right to left, yet they are only the supplements of the versed sines themselves. Now subtract the logarithmic versed sines of such degrees and parts of degrees as are intended to be put on the scale, from the logarithm versed sine of $180^\circ$; then the remainders taken from the foresaid scale of equal parts, and laid successively from the termination of this line, will give the several divisions sought. The following table to every 10 degrees was constructed in the foregoing manner, and are the numbers to be taken from the scale of equal parts, for the degrees they stand against. | Degrees | Supplements of Versed Sines | Degrees | Supplements of Versed Sines | Degrees | Supplements of Versed Sines | |---------|-----------------------------|---------|-----------------------------|---------|-----------------------------| | 180 | 0.00000 | 120 | 0.12494 | 60 | 0.60206 | | 170 | 0.00331 | 110 | 0.17327 | 50 | 0.74810 | | 160 | 0.01330 | 100 | 0.23149 | 40 | 0.93190 | | 150 | 0.03011 | 90 | 0.30103 | 30 | 1.17401 | | 140 | 0.05403 | 80 | 0.38387 | 20 | 1.52066 | | 130 | 0.08545 | 70 | 0.48282 | 10 | 2.21941 | From this table it appears, that the least versed sine, which can be introduced within the length of a double radius, falls between $10^\circ$ and $20^\circ$, where the index changes from 1 to 2; which will happen about $11^\circ 28'$. If a table of logarithm versed sines to $180^\circ$ are wanting, they are easily made by the following rule. Take the logarithm sine of 30 degrees from twice the logarithm sine of $(N)$ any number of degrees; the remainder is the logarithm versed sine of $(2N, \text{or})$ twice those degrees." For it is a well-known goniometrical property, that the sine of any arc $(A)$ is a mean proportional between radius $(R)$ and half the versed sine of twice that arc. Therefore, putting $v$ for the versed sine, and $s$ for the sine; The \( v_2 A = \left( \frac{2 ssA}{R} = ssA \times \frac{2}{R} = ssA \times \frac{1}{5} \right) \) \( ssA \times \frac{1}{5} \); radius being 10. Or the log. \( v_2 A = 2 \log. sA - \log. 5 \). But when radius is 10, the sine of 30° is 5. Therefore the log. \( v_2 A = 2 \log. sA - \log. \text{sine of } 30° \). Most of the writers on this subject give the following rule for laying down the divisions of this line: From the line of logarithmic sines, take the distance between 90° and any arc; that distance being twice repeated, from the termination of the line of versed sines, will give the division for twice the complement of that arc." Thus the distance between 90° and 20° on the sines twice repeated, gives the versed sine of 140°; or twice 70°, the complement of 20°. For the divisions, to be laid on this line, are the differences between the logarithm versed sine of 180°, and the logarithm versed sines of the successive arcs. Now the difference between the logarithm versed sines of 180°, and of any arc \( 2A \), is log. ver. sine 180° - 2 log. fin. \( A + \log. \text{fin. of } 30° \). Or, 10,30103 + 9,69897 — twice log. fin. of \( A \). Or, 20,00000 — twice logarithm fine of \( A \). Or the arithmetical complement of twice logarithm fine of \( A \). That is, the difference between the logarithm versed sine of 180°, and the logarithm versed sine of any arc, is equal to double the arithmetical complement of of the logarithm sine of half that arc, rejecting the indices. But, as these differences give the divisions to the supplements of the real versed sines; therefore the arithmetical complement of the logarithm sine of any arc being doubled, will give the distance of the division for the supplement of twice that arc on the line of versed sines. Thus, for $70^\circ$, the logarithm sine is $9.97299$ The arithmetical complement is $0.02701$ Its double is $0.05402$ Which is the number in the foregoing table standing against $140^\circ$, and is the supplement versed sine of twice $70$ degrees. Now, as the arithmetical complement of the log. sines of arcs, are the distances on the line of sines between $90^\circ$, and the divisions to those arcs; therefore the distances between $90^\circ$ and any arc, being twice repeated, will give the division of the supplemental versed sine to twice the co-sine of that arc. --- XIV. A Letter from Mr. John Dollond to Mr. James Short, F. R. S. concerning an Improvement of refracting Telescopes. S I R. Read March 1, 1753. It is well known, that the perfection of refracting telescopes is very much limited by the aberration of the rays of light from the geometrical focus; which arises from two very different causes; that is, from different degrees