The exterior angle EAB is (III. 17. cor. 2.) equal to BCD, and these angles and their adjacent angles BAD and DCV are bisected by AG and AH, CI and CF; wherefore the right-angled triangles GKA and AOH, IRC and CVF are all similar. Consequently AK: KG:: OH: OA:: CR: RI:: FV: CV, and (V. 19. El.) AK+CR: KG+RI:: OH+FV: OA+CV; whence (V. 13. El.) KQ (AK + CR): OU (KG + RI) : : KQ(OH+FV): OU (OA+CV). But it was shown, that KQ is the excess of the semiperimeter KU above the side CD, that AK+CR, or AK+DU, is the excess of KU above the side AD, that OU is the excess of KU above KO or the side AB, and that OA+CV, or LV + CV, is the excess likewise of the semiperimeter above the side BC. The extreme terms of the analogy are hence the rectangles of the excesses of the semiperimeter above the sides CD and AD, and above the opposite sides AB and BC. It only remains therefore to prove, that the mean terms are each equivalent to the area of the quadrilateral figure.

Now, this quadrilateral figure ABCD is evidently composed of double the triangles HOA, HNB, IQD and IRC, increased or diminished by the double of the rhomboid HOQI, according as the point H lies nearer to the vertex E, or more remote than the point 1: The area of ABCD is therefore equivalent to HO.AB+IQ.DC±(HO+IQ)OQ=HO(AB÷0Q) + IQ (CD÷OQ) = HO.KQ + IQ.OU.

But since the triangles AHN and ISC are similar, and also BMG and DFT, it follows that AN: NH:: IS: SC, and BM: MG :: FT: TD; whence HN; MG :: FT: ÍS, and KG: OH :: IQ: FV; : EK: EO:: EQ; EU, or, by division, KG: HO :: KQ : OU, and IQ.FV :: KQ.OU; wherefore HO.KQ = OU.KG, and the area of ABCD= OU (KG+IQ or IR). Again, IQ.OU=KQ.FV; and consequently the area of the quadrilateral figure is likewise equivalent to KQ (HO+FV). The proposition is therefore established.

For the substance of this demonstration, I am indebted to the talents and assiduity of my ingenious friend Mr Arnott.

The Appendix to the books of Geometry cannot fail, by its novelty and singular beauty, to prove highly interesting. The first part is taken from a scarce tract of Schooten, who was Professor of Mathematics at Leyden, early in the seventeenth century. But the second and most important part is chiefly selected from a most ingenious work of Mascheroni, a celebrated Italian mathematician; which in 1798 was translated into French, under the title of Geometrie du Compas. It will be perceived, however, that I have adapted the arrangement to my own views, and have demonstrated the propositions more strictly in the spirit of the ancient geometry.

NOTES TO TRIGONOMETRY.

1. It was the pursuit of Practical Astronomy that led the Greeks to cultivate Trigonometry. But the Elements of Geometry had been reduced to a systematic form before the foun dation of this science was laid. Aristarchus of Samos, a contem porary of Euclid, and a very ingenious astronomer, attempted to determine the relative distances of the sun and moon by a curious observation; yet the rude and circuitous way which he took, in solving the problem, to show that an arc of three degrees is less than the 18th, and greater than the 20th part of the radius, attests the slow progress of the art of calcula

tion.

About a century afterwards, the great outlines of Trigonometry were traced by Hipparchus of Rhodes, the most original. and powerful genius perhaps of all antiquity, who flourished between the years 125 and 160 before the Christian æra. If we consider the novelty of his views, the extent of his inqui ries, the fecundity of his resources, and the accuracy and immensity of the calculations which he performed with such cumbrous instruments and involved materials—we are filled with wonder and admiration. The main object of his pursuits was the advancement of astronomy. Hipparchus adopted the usual division of the circumference of the circle into 360

degrees; but he divided the radius into 60 equal parts, which he called likewise degrees, and repeated the successive subdivision by 60 for primes, seconds, &c. He derived the rules of Trigonometry from the properties of lines inscribed within the circle, and now termed chords, which he measured in sexagesimal parts of the radius. He had composed a work in twelve books expressly on the calculation of chords, and appears to have computed these important lines to every half degree of the semicircumference. Unfortunately some fragments only of his labours are preserved.

Trigonometry, in this shape, remained stationary during the space of 300 years, till Ptolemy, who cultivated Astronomy and Geography in the Museum of Alexandria with indefatigable ardour, introduced a few subordinate improvements. In his Analemma, which deduces the construction of sun-dials from the orthographic projection of the sphere, this celebrated author, who seems to have borrowed liberally from Hipparchus, employs half-chords, instead of chords, in his geometrical delineations; and, had he pursued that simple idea, he might have anticipated the use of sines, the introduction of which was reserved for the Arabians.

Ptolemy recomputed the chords to every two degrees of the semicircumference, and, in his Almagest, or Mıyaλn Zvilağıs, he has explained distinctly the mode of proceeding. On this occasion, he appears first to have brought into notice the beautiful theorem which forms the 20th proposition of the sixth Book of the Elements of Geometry.

An an object of considerable interest, I here subjoin Ptolemy's table of chords, or, as he styles it, the Canon of Inscribed Lines-Κανονιον τῶν ἐν Κυκλῳ εὐθείων. It will be found to be far more precise and accurate than we should have expected; and it shows the wonderful address and patience which the Greeks exercised in performing calculations with their complicated system of arithmetical notation. For the sake of instituting a comparison, I have converted the sexagesimals into decimals, and have likewise annexed, from the more elaborate modern tables, the chords, expressed sexagesimally. The coincidence will appear very remarkable, seldom differing by unit in the last place.

TABLE OF CHORDS to every two Degrees of the Semicircumference, in Sexagesimals, and the same as given by Ptolemy, and likewise converted into Decimals.

CHORDS.

Sexagesimal Scale.

By Modern Tables. By Ptolemy. By Ptolemy.

To understand the ancient writers on Astronomy and Trigonometry, it is necessary to be familiar with the methods of

4'

converting sexagesi- 37° 4′ 55"

Or more con

cisely, thus:

lemy assigns for the

10

chord of 36°, or the

2.24 26 40

10

side of the inscribed

4.0 26 40

decagon: The con

version may be performed either by a successive multiplication by 10, or by an ascending division by 60.

Again, let it be required to express sexagesimally the chord of 80°, or 1.2855752: The operation will consist in a repeated multiplication by 60.

The result is 77° 8' 4"; but Ptolemy makes the last figure to be 5".

Albatenius or Geber, the son of Mahomet, an Arabian Prince, who flourished about the year 880 of the Christian æra, wrote a system of Astronomy, in which he improved on the Almagest of Ptolemy. To simplify the calculations of Trigonometry, he substituted for chords, their halves, which, in the Arabic language, he named Gib, to signify folded or doubled up. This word, appropriated to the semichord, was afterwards translated into Latin by the term sinus, from which comes the modern denomination sine. Albatenius likewise introduced versed sines, and thereby improved the rules of Trigonometry. He made even a decided step towards the formation of tangents; for, in treating of dialling, he computed a table of the lengths of the shadows of the vertical style, though