A Breviate of Monsieur Picarts Account of the Measure of the Earth

A Breviate of Monsieur Picarts Account of the Measure of the Earth. This Account hath been printed about two years since, in French; but very few Copies of it being come abroad, (for what reasons is hard to divine;) it will be no wonder, that all this while we have been silent of it. Having at length met with an Extract thereof, and been often desired to impart it to the Curious; we shall no longer resist those desires, but faithfully communicate in this Tract what we have received upon this Argument from a good hand. The Author then, whose name is not prefixed to the Book (though generally 'tis thought to be the Intelligent and Learned Picartus, an Eminent member of the Royal Academy of the Sciences at Paris,) divides his Treatise into 13 Articles; of which we shall first of all represent the sum, as 'twere, in one view; and then, for the satisfaction of the more curious, deliver the Breviat of every Article. The Sum then of the whole amounts, in short, to this; That the French Author hath found 5760 toises or fathoms for one degree, that is, 28 leagues and 60 toises; which being multiplied by 360 (the number of the degrees) makes 10270 leagues and 1600 toises, reckoning 2000 toises to a league, or 2400 paces, 5 foot to a pace. The Method employ'd by him hath been, To measure on a plain and straight ground a space of 5663 toises, to serve for the first basis to divers Triangles, by which he hath concluded the Length of a Meridian line to be equivalent to a degree. That which is remarkable for the certainty of this Observation, is, 1. That no body, we know of, hath ever measured so great a basis; the greatest of the former Observations having been but of a 1000 toises. 2. That here have been employed, for taking the Angles of position, very accurate Instruments, and Telescopic Sights instead of common ones; all described in the said Book: of which we shall now proceed to deliver the import of every Article. In the first then, he begins with a Preamble, shewing, that this Problem concerning the Just Dimensions of the Circumference of the Earth is no New thing, but hath been the Inquiry of several Ages, in which Princes have been curious, and Learn'd men encouraged to the search and clearing of this Difficulty. And to this purpose he alledges a passage out of Abulfeda, to this effect, that Almamon, a Prince of the Arabes, desirous to know, what the True measure of a Celestial Degree might be upon Earth, commanded the Experiments to be made in the Plains of Sanjar; where a Station being chosen, and thence Troops of Horsemen let out, that went in a straight line, till one of them had raised a degree of Latitude, and the other had deprest it; at the end of both their marches, they who raised it, counted 563 miles, and they who deprest it, reckon'd 56 miles just. This Observation can instruct us but very little, because we know not justly, of what length these miles were. Then, the Author observes, that the Ancient computations of miles for a Degree run always upon the decrease; so as Aristotle counting to a Degree 1111 stadia, after him Eratosthenes counted but 700; Posidonius but 666; Ptolemy but 500. Nor do these Observations teach us any thing certain, because the precise length of these stadia is unknown to us; and they were also different among themselves; the stadia of Alexandria differing from those of Greece. At last Fernelius brought it to 5674.6 Toises or Fathoms of Paris, each of which is equal to 6 Parisian feet; Snellius, to 55021. In the second Article, he judges Snellius his way of measuring to be the most artificial; which was by a Scale of Triangles. But in one thing he esteems it deficient, which is, that Snellius took his Object only by Pinnules, or Common Sights, which do not so distinctly point it out. In the third Article he begins to speak of his own Method, and its exactness, and saith, That, when the resolution was taken of See Fig. 1. Measuring a Degree, he chose his Meridian, out of which Tab. 1. he intended to take his Measure, between Sourdon in Picardy, and Malvoyfin, which is upon the confines of the Gaistinois and Hurepois. To attain the exact Measure of this Arch of the Meridian, lying between Sourdon and Malvoyfin, he saith, he actually measur'd a way that lay very straight, between Villejuifve and Ivoisy, viz. A.B. And he began to measure from the middle of a Mill at Villejuifve, and continued till he came to the Pavillon of Ivoisy, and found the distance between these two termines, in going forward, to be 5662 toises and 5 feet, and in comming back, 5663 toises and 1 foot; which being measured with great exactness, he stated the distance between these two places, in round reckoning, 5663 toises. The The Instrument he measured with, was Pikes joined together at their ends by a screw, which measure was 4 toises long: This he applied along a cord stretched horizontally, and at the end of every such Pike had a stake; of which stakes he had 10 in all. This distance of 5663 toises was the Base of the first Triangle, upon which the measure of all the depending scale was formed. Here in Art. 4. he takes occasion to discourse of Measures in general, and saith, That a Pendul vibrating a second of time, computed according to the Mean motion of the Sun, is 36 inches and 8½ lines of the Chastelet of Paris measure. And he esteems, that this Measure may probably serve in all Countries, because the same Length of a Pendul served for a Second both at the Hague and Paris; whence he conjectures, the same may serve also in other Latitudes. Whereupon he infers, that if one had a mind to constitute an Universal Measure, which might be common to all Countries, it might be thus made, viz. Call this Pendul for seconds (of 36 inches and 8½ lines) the Astronomical radius; the ¼ of this radius the Universal foot; the double of which radius might be called the Universal Toise or Fathom, which would be to the Parisian Toise as 881. to 864; the Quadruple might be called the Universal Perch, which is equal to the length of a Pendul for two seconds. In a word, the Universal Mile might contain a 1000 of these Perches. The Instrument, (in Art. 5.) wherewith the Angles were taken in the Mensuration of the Triangles, was a Quadrant of 38 Inches radius, furnish't with Telescopical-glasses, the better to point out the Objects: Which Instrument, he saith, never miss'd a minute in taking an Angle; sometimes it came within five seconds. But to proceed; In the sixth Article he relates the Manner of taking the Distance between Sourdon and Malvoyzin, together with the Triangles, and the Stations from whence he observ'd his Angles. This distance is 68343 toises and 2 feet. The Base, which he actually measur'd, as we said above, was AE, the high way lying between Villejuifve and Ivoisy, which he found, (as hath been already intimated) equal to 5663 toises of Paris. And from this Base he deduced the measure of all the 13 Triangles, viz. In the first Triangle AEC, to find the side AC, BC. \[ \begin{align*} \text{Angles} & \\ \text{CAB} &= 54^\circ 4' 35'' \\ \text{ABC} &= 95^\circ 6' 55'' \\ \text{ACB} &= 30^\circ 48' 30'' \end{align*} \] Toises. Feet. The Side AB is 5663. of equal measure. Hence AC is 11012. 5. BC 8954. 0. In the second Triangle ADC, to find DC and AD. \[ \begin{align*} \text{Angles} & \\ \text{DAC} &= 77^\circ 25' 50'' \\ \text{ADC} &= 55^\circ 0' 10'' \\ \text{ACD} &= 47^\circ 34' 0'' \end{align*} \] Toises. Feet. The Side AC is 11012. 5. Hence DC 13121. 3. AD 9922. 2. In the third Triangle DEC, to find DE, CE. \[ \begin{align*} \text{Angles} & \\ \text{DEC} &= 74^\circ 9' 30'' \\ \text{DCE} &= 40^\circ 34' 0'' \\ \text{CDE} &= 65^\circ 16' 30'' \end{align*} \] Toises. Feet. The Side DC 13121. 3. Hence DE 8870. 3. CE 12389. 3. In the fourth Triangle DCF, to find DF. \[ \begin{align*} \text{Angles} & \\ \text{DCF} &= 113^\circ 47' 40'' \\ \text{DFC} &= 33^\circ 40' 0'' \\ \text{FDC} &= 32^\circ 32' 20'' \end{align*} \] Toises. Feet. The Side DC 13121. 3. Hence DF 21658. c. In the fifth Triangle DFG, to find DG, FG. \[ \begin{align*} \text{Angles} & \\ \text{DFG} &= 92^\circ 5' 20'' \\ \text{DGF} &= 57^\circ 34' 0'' \\ \text{GDF} &= 30^\circ 20' 40'' \end{align*} \] Toises. Feet. Side DF 21658. c. Hence DG 25643. 0. FG 12963. 3. In the sixth Triangle, GDE, to find GE. The Angle GDE = $128^\circ 9' 30''$. Toises, Feet. | The Sides | SDG | DE | |-----------|-----|----| | | 25643. 0 | 8870. 3 | Hence GE = 31897. 0. So then, the Line of Distance between Malvoisin and Sourdon being divided into three parts, viz., EG, GI, IN, the part EG is already found. In the seventh Triangle FGH, to find GH. Angles | FGH = $39^\circ 51' 0''$ | | FHG = $91^\circ 46' 20''$ | | HFG = $48^\circ 22' 30''$ | T. F. The Side FG = 12963. 3. Hence GH = 9695. 0. In the eighth Triangle GHI, to find GI, IH. Angles | GHI = $55^\circ 58' 0''$ | | GIH = $27^\circ 14' 0''$ | | IGH = $96^\circ 48' 0''$ | T. The Side GH = 9695. Hence GI = 17557. HI = 21037. Thus the Second part of the Three, viz., GI, is found. In the ninth Triangle HIK, to find IK. Angles | HIK = $65^\circ 46' 0''$ | | HKI = $80^\circ 59' 40''$ | | KHI = $33^\circ 14' 20''$ | T. The Side HI = 21043. Hence IK = 11678. In the tenth Triangle IKL, to find KL, IL. Angles | LIK = $58^\circ 31' 50''$ | | IKL = $58^\circ 31' 0''$ | T. F. The Side IK = 11683. 0. Hence KL = 11188. 2. IL = 11186. 4. In the eleventh Triangle KLM, to find LM. Angles \( \begin{align*} \text{LKM} &= 28^\circ 52' 30'' \\ \text{KML} &= 63^\circ 31' 0'' \end{align*} \) The Side KL = 11188. 2. Hence LM = 6036. 2. In the twelfth Triangle LMN, to find LN: Angles \( \begin{align*} \text{LMN} &= 60^\circ 38' 0'' \\ \text{MNL} &= 29^\circ 28' 20'' \end{align*} \) The Side LM = 6036. 2. Hence LN = 10691. 0. In the thirteenth Triangle ILN, to find NI. The sum of the Angl. IKL, KLM, MLN, taken from 360, there remains Angle ILN = 119° 32' 40''. The Sides \( \begin{align*} \text{ILN} &= 10691. 0 \\ \text{IL} &= 11186. 4 \end{align*} \) Hence IN = 18905. 0. Thus, the Line of Distance, EN, being, as hath been said, divided into three unequal parts, EG, GI, IN, the measures of all three are found by this Scale of Triangles. Now then, reassuming what hath been already discover'd by the help of these Triangles, and finding, that EG was in length 31897. GI = 17557. IN = 18905. These added together, make the length of EN, which is the Line of Distance between Malvoysin and Sourdon, viz. 68359. Now to continue this measure from Sourdon to Amiens, (which is the business of the seventh Article, undertaken to the end that Ferneilius his account might be liquidated, whether it were true or no;) you must, for the attaining it, make use of the Diagram of Fig. 2; where R. stands for the Steeple of St. Peters in Montdidier; T. is a Tree upon the Hill of Mareuil; V. is the Lantern of Nostre Dame of Amiens. To find the distance NV, you must look back upon NLM, the last Triangle of Fig. 1, and see, how it is disposed in Fig. 3; where in the Triangle LMR, The Angles \( \sum LMR = 58^\circ 21' 50'' \). \( \sum MRL = 68^\circ 52' 30'' \). T. F. The Side LM 6037. 0. Hence LR 5510. 3. In the Triangle NRL. The Angles \( \sum NRL = 115^\circ 1' 30'' \). \( \sum RNL = 27^\circ 50' 30'' \). T. F. The Side LR 5510. 3. Hence NR 7122. 2. Go on to Fig. 2. in the Triangle NRT. The Angles \( \sum NTR = 72^\circ 25' 40'' \). \( \sum TNR = 67^\circ 21' 40'' \). T. F. The Side NR 7122. 2. Hence NT 4822. 4. Finally in the Triangle NTV, The Angles \( \sum NTV = 83^\circ 58' 40'' \). \( \sum TNV = 70^\circ 34' 30'' \). T. F. The Side NT 4822. 4. Hence NV 11161. 45 which was sought. Now, adding the Dist. between Malvoisin & Sourdon, viz. 68359. 0. to the distance between Sourdon and Amiens 11161. 4. The whole will be the dist. between Malvoisin & Amiens 79520. 4. Having thus measured the particular distances between Malvoisin, Mareuil, Sourdon, and Amiens, he proceeds to examine, in the eight Article, the Position of each of these Lines of distance in respect of the Meridian, or to deduce the Length of the Meridian intercepted between the Parallels of Malvoisin and Amiens: Which was thus done; In Septemb. 1669. he went to the Hill of Mareuil, and from the top of it, which is mark't with G in Fig. 1. (from whence one can discern Clermont on one side, at I, and Malvoisin on the other side, at E.) he took the Meridian, and with a Quadrant took the Angles of Declination from this Meridian. The manner he relates at length; the result whereof is, That by these Observations he found, The Angle EG in Fig. 1, which is the Declination of EG from the Meridian westward The Angle GIθ, which is the Declination of GI from the Meridian Eastward The Angle INV, which is the Declination of IN from the Meridian Eastward The Angle VNB in Fig. 2, which is the Declination of NV from the Meridian Westward So that in all these 4 Triangles, EG, GIθ, INV, VNB, you have two Angles known (for the Angles at ε, at θ, at V, at β, are right,) and a side, viz. EG, GI, IN, NV. where he concludes, The length of the Meridian G ε to be 31894. 0. of the Meridian I θ 17560. 3. of the Meridian NV 18893. 3. of the Meridian N β 10559. 3. And hence the length of the whole Meridian ε between the Parallels of Malvoisin and Amiens to be 78907. 3. Here he casts in an Objection, and saith, that these Lines, which make up the Meridian, are not, in a strict sense, a Curve, but in reality the side of a Polygone circumscribed about the Circumference of the Earth. But, for answer to this, he affirms the Difference between those Lines and a true Curve to be but 3 foot per degree, which he saith is scarce worth taking notice of. This he proveth afterwards, where he makes the Table, in which he calculates, what difference there is between the real Level and the apparent. To this he subjoyns a Note, importing, that though he took these Meridians, for greater exactness, with a Quadrant; yet he omitted not to use a Compass, whose Declination to the Westward, he saith, in the Year 1670, towards the end of the Summer, he found 1° 30'. Whereas A. 1666. he observed very little variation, if any at all. But A. 1664. it varied Eastward 0° 40'. Here he makes a pretty Note, telling us, that the Difference of Variation in a years time amounts to 0° 20'. The Length of the Meridian between Malvoisin and Amiens being thus stated, his next business is, in the ninth Article, to enquire, What answers to it in the Heavens, comparing those Meridian distances, already measur'd, with Minutes and Seconds there: which were were taken by the help of an Instrument, whose Limb was an Arch of \( \frac{1}{20} \) of a Circle of 10 foot radius; whereof he gives the Figure, and his manner of rectifying any Errors, which in using it might deceive him. In the tenth Article he relates, that the knee of Cassiopeia was the Starr he pitch't on, from whence to measure the Minutes and Seconds of a Degree in the Heavens; adding the reasons, why he chose that Starr. In the eleventh he gives the resolution of the thing in Question, that is to say, How many Toises or Fathoms, Parisian measure, answer a Degree of the Circumference of the Earth; as for instance, the Difference of Latitude between Malvoisin and Sourdon is found, by Observations made in the Heavens, to be \( 1^\circ 11'57'' \). Between Malvoisin and Amiens \( 1^\circ 22'55'' \). Now, the Meridian distance between Malvoisin and Sourdon, calculated from Measures taken upon Earth, was, Toises. Feet. as may be seen above 3 \( 60430.3 \). Whence 'tis concluded, that 57064 Toises and 3 feet, or, in a round number, 57060 Toises are equal to a Degree. Which if you would reduce to Universal Measure, you are to remember, that the Universal toise is to the Parisian, as 881 to 864: Whence one Degree is equal to 55959 Toises Universelles. The Reduction of which to the measures of other Countries is this; Suppose the Paris foot to consist of | Measure | Parts | |----------------------------------|-------| | The Rhynland (or Leyden) foot | 1440. | | The London-foot | 1330. | | The Bolonian-foot | 1350. | | The Braccia of Florence | 1686. | | Hence a Degree in a grand Circle of the Earth, according to the Measures of different Countries, is, | Measure | Parts | |----------------------------------|-------| | Toises du Chastelet de Paris | 57060.| | Pas de Bologna | 58481.| | Verges du Rhin de 12 pieds chacune| 29556.| | Lieues Parisiennes de 2000 Toises chacune | 28\(\frac{1}{4}\).| | Lieues moyennes de France d'environ 2282 Toises | 25.| | Lieues de marine, de 2853 Toises | 20.| | Milles d'Angleterre, de 5000 pieds chacune | 73\(\frac{1}{3}\).| | Milles de Florence, de 3000 braies | 63\(\frac{1}{3}\).| Hence Hence the Circumference of the Earth, In Parisian Toises or Fathoms 20541600. In Leagues of which 25 make a Degree 9000. In Marine Leagues 7200. The Diameter of the Earth is, In Parisian Toises 6538594. In Leagues of 25 to a Degree 2464\(\frac{1}{2}\). In Marine Leagues 2291\(\frac{1}{2}\). He also gives a Table, shewing the Correspondent value in measure to the parts of a Degree: E. g. Min. Toises. Second. Toises. 1 = 951 | 1 = 16. 2 = 1902 | 2 = 32. 60 = 57060 | 60 = 951. After this follows a Table of the Difference of Latitude, which is Between Malvoisin and the Observatoire of Paris 19'. 22". Between Malvoisin and Nostre Dame de Paris 20. 22. Between Malvoisin and Mareuil 33. 32. Clermont 52. 0. Sourdon 71. 52. Nostre Dame d' Amiens 82. 58. Between Nostre Dame of Paris and of Amiens 62. 36. Then follows a Table of Elevations of the Pole of several places, as | In the Garden of the R. Academy at Paris is, | | At Nostre Dame de Paris | | At St. Jaques dela Boucherie | | At Malvoisin | | At the Observatoir of Paris | | At Mareuil | | At Clermont | | At Sourdon | | At Nostre Dame d'Amiens | The Elevation of the Pole 48°. 58'. 0". 48. 52. 10. 48. 52. 20. 48. 31. 48. 48. 51. 10. 49. 5. 20. 49. 22. 48. 49. 43. 40. 49. 54. 46. As to Differences of Longitudes; Sourdon \{ Amiens \} \{ 0°. 5′. 54″. \} Clermont \{ Sourdon \} \{ 0. 1. 9. \} Mareuil more Easterly then Clermont \{ by \} \{ 0. 0. 34. \} Mareuil Malvoisin \{ 0. 0. 20. \} Mareuil Paris \{ 0. 4. 37. \} So much of the eleventh Article. The twelfth is framed upon an Objection, that might be made, viz. Whether the Measure is the same taken at Paris, with that which is taken upon a Level by the Sea-side. Here he computes upon the fall of the River Seine, and judgeth the place where he measured to be raised above the Sea not more than 80 Toises; and concludes the Difference between measuring at Paris and by the Sea not above 8 feet per degree. Where he makes a Table of Levels; describes an Instrument to take Levels with; discourses of Refractions, and how to correct them. In the thirteenth Article he examins several opinions, different from his, concerning this subject; as of Fernelius, Snellius, and Riccioli; and points at the occasions of their respective mistakes; delivering withal the Differences of their Measures from his. Of the three, Fernelius comes the highest; which M. Picart imputes to meer chance, since he used not half the exactness in observing that Snellius did. Snellius his difference from accurateness he attributes, 1. To too small a base, he took to measure, and to too small triangles, which he was forced to take afterwards: 2. To the want of so good Instruments, as were employed in these Observations. To adde something of the three Figures; they represent the Connexion of Triangles, by which our Author measur'd the Distance from Malvoisin to Sourdon, and from Sourdon to Amiens: From which measure he concluded, what the just length of a Degree might be, reduced to the Parisian Toise. Concerning which Triangles nothing more needs to be added, but only a fuller Explication of what the Letters in them do stand for; viz. A. the middle-point at the Mill of Villejuifve. B. the nearest corner of the Pavillon of Ivosly. C. the top of the Steeple of Brie Compte Robert. D. the middle of the Tower of Montleher. E. the top of the Pavillon of Malvoisin. F. a pole placed for this purpose on the ruins of the Tower of Montjay, with a lock of hay put upon it, that it might be seen at a greater distance. G. the middle of the Hummock of Mareuil, where it was requisite to have a fire made, to distinguish it at a distance. H. the middle of the great Oval Pavillon of the Castle of Dammartin. I. the Tower of St. Sampson in Clermont. K. the Mill of Jonquieres near Compiegne. L. the Tower of Coyvrel. M. a little Tree on the hill of Boulogne near Montdidier. N. the Tower of Sourdon. O. a little forked Tree upon the point of the Griffon near Villerneuve St. George. P. the Tower of Montmartre. Q. the Tower near St. Christopher at Senlis. Thus we have given you, we hope, some satisfaction as to this point, at least as to the material parts of it. As to all the particular niceties, (which it would be too tedious to describe) the Book itself, which surely some time or other will come abroad, may render that satisfaction compleat. Mean time, I would by no means, that this should put a stop to the Ingenuity and Industry of our Philosophical Friends here in England, or deprive either them of the pleasure of comparing their exactness with that of M. Picarts, or the world of the advantage of having so important a Problem resolved by divers Artists in different Countries, by different ways; that so, the whole coming to be reflected upon, one may be able to conclude from the accurateness of the Observers, who they are that are come the nearest to truth in their Observations. An Extract of the French Journal des Scavans, concerning a New Invention of Monsieur Christian Hugens de Zulichem, of very exact and portable Watches. The Watches of this Invention being made in small, shall serve for very exact Pocket-watches, and when made greater, of M. Hugens.