TO THE

FORTY-EIGHTH VOLUME

OF THE

MONTHLY REVIEW

ENLARGED.

FOREIGN LITERATURE.

Mémoires de l'Institut, &c. Memoirs of the National Institute of Arts and Sciences. Vol. V. 4to. Paris. Imported by De Boffe.

WE

E resume our analysis of this livraison of the proceedings of this learned body, by directing our attention to the volume which relates to the

MATHEMATICAL and PHYSICAL SCIENCES; and we shall divide its contents under two corresponding heads, beginning with the

MATHEMATICAL and ASTRONOMICAL PAPERS.

HISTORY. Report made to the Mathematical and Physical Class, concerning the Astronomical and Nautical Observations of Joseph Joachim de Ferrer. By M. LEVÊQUE.-The observations here mentioned form the basis of certain geographical determinations made in South America and the Azores. M. Ferrer having very properly described the Instruments, and the methods and formulas of calculation which he employed, together with the precautions and artifices suggested or made necessary by local circumstances, the Commission, in whose name M. LEVÊQUE speaks, turned their attention to these points: for thus only they could be enabled to judge of the accuracy of M. Ferrer's geographical determinations. The Instruments employed by that gentleman were of English manufacture, and by the best Artists: except in one instance, the Commission approves of his methods; the formulas of calculation were such AFP. REV. VOL. XLVII.

G g

as are generally known and used, and no objection therefore could be made against these; and, altogether, it appears that much benefit has been conferred on Geography by M. Ferrer. In the present report, several particular Geographical Determinations are put down; and we hope that the whole will be made public, in order that our Maps and Charts may speedily receive the necessary corrections.

Report relative to a new system of constructing Masts for Vessels. By M. LEVÊQUE.-The new invention here reported is that of a Mast-maker at Rotterdam; by the adoption of which, Masts are to be made lighter and more easy of repair. The Class, to whose judgment the Invention was submitted, approved the design; and the Reporter introduces into his paper some interesting information concerning several problems in Nautical Mechanics.

Report made to the Mathematical and Physical Class. By M. CAMUS. This paper refers to an examination ordered to be instituted respecting the capacity of the Antient Paris Pint. In the new measures, this pint is fixed at 46.95 cubic inches but it was contended that the capacity ought to be 48. This latter opinion originated, it appears, from the existence of pint measures used in 1747, and which the Parliament had proscribed in 1750.

Report on a Memoir of M. Gail, intitled Description of an Astrolabe by Synesius. By M. DELAMBRE.

MEMOIR. On the Stereographic Projection; by the same. We have ranged these two papers together, because the memoir was drawn up in consequence of the Report. The memoir of Gail is almost a complete translation of Synesius's letter to Pæonius, which accompanied the present of a silver Astrolabe; and it appears, by the evidence of this letter, that Hipparchus was the original Inventor of the Planisphere, and not Ptolemy. This latter Astronomer was probably not fully acquainted with all the properties of the stereographic projection for instance, he did not know that all circles of the sphere, excepting those of which the plane passes through the eye, are in the stereographic projection always projected into circles; and this remarkable property was, in fact, for a long period, within the reach of Mathematicians, without their availing themselves of it. It is demonstrated in Apollonius, that the sub-contrary section of a cone is, like the base, a circle. The step to be made, therefore, in order to prove the projection of a circle to be a circle, was that the plane of projection forms a sub-contrary Section in every cone, the vertex of which is the eye, and the base a circle of the sphere.

Easy

as

as this step is, it was not achieved till fifteen hundred years after the time of Hipparchus.

One remarkable property of the Stereographic projection has been already mentioned: another, equally curious, unknown to Ptolemy, is that the circles of projection intersect each other in the same angle as the circles on the sphere. The proof of this property is the object of M. DELAMBRE's memoir; and it is effected with great conciseness, and with what may be called mathematical elegance. In fact, he deduces the proof from two expressions; one, that of the distance of the centre of the sphere from the projected pole of the projected circle; the other, that of the radius of the projected circle. If P be the pole, PE the arc drawn from P to the circumference of the circle to be projected, and if AP be the distance of P from the pole of projection, then the expression for the distance of the centre of the sphere from the pole of the projected circle =

circle =

sin. AP

cos. AP+cos. PE and radius of projected

sin. PE

cos. AP+cos. PE'

M. DELAMBRE subjoins a geometrical demonstration of the property of the projection, and applies his formulas to the construction of maps which represent the earth projected. His memoir also contains the solution of all the problems which Synesius says he had resolved.

In the report and in the connected memoir, considerable doubts are expressed whether Ptolemy was the author of the Treatise of the Planisphere. Of this treatise we have only a Latin Translation made from the Arabic.

We recommend the perusal of this paper to all lovers of mathematical science.

Notice respecting the great Logarithmic and Trigonometrical Tables calculated at the Board of Registry of Lands, under the direction of M. PRONY. By M. PRONY.

Report relative to the grand Trigonometrical Tables. By M. DELAMBRE.

The decimal division of the circle rendered the construction of new tables necessary; and ten years ago, M. PRONY was appointed to superintend and direct their formation. He was desired to select his coadjutors, not only for the obvious purpose of rendering the tables as exact as possible, but, to use his own words, à en faire le monument de calcul le plus vaste et le plus imposant qui eût jamais été executé ou même conçu.'

The principle of the division of labour was applied to the construction of these tables. The calculators were divided Gg 2

into

into three sets; the first consisting of the most able mathematicians, directing and superintending the analytical part: the second, of seven or eight skilful computists in arithmetic as well as in analysis, employed in deducing numbers from the general formulas: the third set, (and the largest, their number being from 60 to 80,) consisted of persons who were only required to add and subtract.

We observe that the method of calculating these tables differs, in some respects, from that which has been usually employed. In the present paper, the method is generally described, and we shall endeavour to make it clear to our readers by a few particulars.

Suppose to be a function of x, and successive values of to be v, v2 V3 บ --V

2

Let the difference between two successive values be denoted by A v

Hence, Av, Av &c. being known, v, may be determined. Suppose the Logarithms of 1001, 1002, 1003, &c. are to be determined: the logarithm of 1000 is 3v, of 1001 = ~,, of 1002 = 2 &c. Now generally log. (x+i) log. x+m

In the present instance, x = 1000, i = 1, m =.43429 &c.

..log. 1001 or v + Av = 3.434147 &c.

Again, v2 = "11 + A v1 = v1 + Av + Av2.

Το

To the logarithm of 1001, then, computed as above, add the quantity Av+ Av2 computed from the preceding forms; and we obtain the logarithm of 1002.

Again, v12 + Av2 = v2 + A v1 + A2 v,

= v2+ Av + 24 2v + A3

2

To the log. of 1002, add the quantity Av+2Av + A'v computed as above, and we have the logarithm of 1003.

By this method, it is clear, the logarithms of numbers from 1000 to 1010, for instance, may be computed by simple additions. This number may serve as a new point of departure: thus, in preceding formulas, for x put 1010, and calculate the quantities Av, Av, A3v, &c. and then the logarithms of numbers from 1010 to 1020, by a process the same as that which we have shewn, may be obtained. For exactness, and for the sake of verifying the computations, the logarithm of 1010 ought to be calculated from the logarithmic series; and if the compu tation has been rightly conducted, the logarithm of 1010 so computed ought to agree with the logarithm obtained by the successive addition of the differences Av, Av, &c. to the logarithms of 1001, 1002, &c.

The process is similar for the calculation of sines. By employing the trigonometrical formulas, Av, ▲1v, &c. must be computed; and then successively v, v, v, &c. thus,

v1 = sin. (x + i) = sin. x .cos. i + cos. x .sin. i. .. Av=sin. x

(cos. ¿-1) + cos. x sin. i: but cos. i—1 — — 2

.. Av or a sin. x cos. x. sin. i—sin. x .2

2

(sin.
sin. -)*

2

Similarly, we may compute A3v— — 4. sin. (x + i) ( sin. — )*

Av &c.

Hence it is evident that we may compute, as in logarithms, by continual addition, the sines of arcs ascending in arithmetic progression: thus v2 = v. + Av: let the difference of the arcs be 1 minute, i1': then sin. (x+2') = sin. (x+1') + A .sin. (x + 1') = (sin. + 1') + A sin. x + A2 sin. x = sin,

2

(*+ 1') + { sin. (x+1')—sin. x } — sin. (*+ 1') (2 sin. 301)* }

So also,

sin. (x+3')=sin. (x+2') + {sin. (x-1-2′)—sin. (x+1')}

sin. (x+2)/2.sin. 302)

The sines of arcs thus computed by successive additions were compared, at certain intervals, with sines immediately

Gg3

deduced