EXAMPLES.-1. The diameter of a sphere is 3 feet; required

its solidity?

First, 3 x3.1415927=9.4247781-circumf. of the cylin

Thirdly, 7.0685836 × 3=21.2057508=the solidity of the cylinder. Art. 292.

Lastly,

of 21.2057508=14.1371672 cubic feet the solidity of the sphere.

2. The diameter of a sphere is 17 inches; required its solidity? Ans. 1.48868 cubic feet.

3. If the earth be a perfect sphere of 8000 miles diameter, how many cubic miles of matter does it contain?

PART IX.

TRIGONOMETRY,

HISTORICAL INTRODUCTION.

TRIGONOMETRY is a science which teaches how to determine the sides and angles of triangles, by means of the relations and properties of certain right lines drawn in and about the circle; it is divided into two kinds, plane and spherical, the former of which applies to the computation of plane rectilineal triangles, and the latter to triangles formed by the intersections of great circles, on the surface of a sphere.

This science is justly considered as an important link connecting theoretical Geometry with practical utility, and making the former conducive, and subservient to the latter. Geography, Astronomy, Dialling, Navigation, Surveying, Mensuration, Fortification, &c. are indebted to it, if not for their existence, at least for their distinguishing perfections; and there is scarcely any branch of Natural Philosophy, which can be successfully cultivated without, the assistance of Trigonometry.

We are in possession of no documents that will warrant us even to guess at the period when Trigonometry took its rise; but there can be do doubt that it must have been invented not very long after the flood. The earliest inhabitants of Chaldæa and Egypt were acquainted with Astronomy, which

The name is derived from rgus three, yovos a corner, and μsrgew to measure. The objects of Trigonometry are the sides and angles only, whatever respects the areas of triangles belongs to Geometry.

(admitting it to have been at that time merely an art, and in its rudest state) would still require the aid of some method similar to Trigonometry to make it of any benefit to mankind.

We may reasonably suppose that the ancient Greeks cultivated Trigonometry, in common with Geometry and Astronomy; but none of their writings on the subject have been preserved. Theon, in his Commentary on Ptolemy's Almagest, mentions a work consisting of twelve books on the chords of circular arcs, written by Hipparchus, an Astronomer of Rhodes, A. C. 130. This work is believed by the learned to have been a treatise on the ancient Trigo

b Theon, a respectable mathematician and philosopher, and president of the Alexandrian school, flourished A. D. 370. He was not more famous for his acquirements in science, than for his veneration of the DEITY, and his firm belief in the constant superintendence of divine providence; he recommends meditation on the presence of God, as the most delightful and useful employment, and proposed, that in order to deter the profligate from committing crimes, there should be written at the corner of every street, REMEMBER GOD SEES THEE, O SINNER. Dr. Simson, in his notes on the Elements of Euclid, has ascribed most of the faults in that book to Theon, without mentioning on what authority he has done so.

C

Hipparchus was born at Nice, in Bithynia: here, and afterwards at Rhodes and Alexandria, his astronomical observations were made. He discovered that the interval between the vernal and autumnal equinox is longer by 7 days than that between the autumnal and vernal; he was the first who arranged the stars into 49 constellations, and determined their longitudes and apparent magnitudes; and his labours in this respect were considered so valuable, that Ptolemy has inserted his catalogue of the fixed stars in his Almagest, where it is still preserved. He also discovered the precession of the equinoxes, and the parallax of the planets; and, after the example of Thales, and Sulpicius Gallus, foretold the exact time of eclipses, of which he made a calculation for 600 years. He determined the latitude and longitude, and fixed the first meridian at the Fortunatæ Insulæ, or Canary Islands; in which particular he has been followed by most succeeding geographers. Astronomy is particularly indebted to him for collecting the detached and scattered principles and observations of his predecessors, arranging them in a system; thereby laying that rational and solid foundation, upon which succeeding astronomers have built a most sublime and magnificent superstructure. Of the several works said to have been written by him, his Commentary on the Phoenomena of Aratus is the only one that remains.

nometry, and is the most ancient on that subject of which we have any account.

The Spherics of Theodosius is the earliest work on Trigonometry at present known. It was written about 80 years before Christ, and consists of three books, " containing a variety of the most necessary and useful propositions relating to the sphere, arranged and demonstrated with great perspicuity and elegance, after the manner of Euclid's Elements.”

We are in possession of three books on spherical triangles by Menelaus. He is considered as the next Greek writer who treated expressly on the subject, and lived about a hundred after Christ. This work of Menelaus was greatly

years

* Theodosius was a native of Tripoli, in Bithynia; and, according to Strabo, excelled in mathematical knowledge. The work above-mentioned consists of three books; the first of which contains 22 propositions, the second 23, and the third 14. It was translated into Arabic, and afterwards from the Arabic into Latin, and published at Venice; but the Arabic edition being very defective, a complete edition was obtained by Jean Pena, Regius Professor of Astronomy at Paris, and published there in Greek and Latin, A. D. 1558. Long before this time, a good Latin translation of the work had been made by Vitellio, a respectable Polish mathematician of the 13th century, and the first of the moderns who wrote to good purpose on optics. The Spherics of Theodosius have been enriched with notes, commentaries, and illustrations, by Clavius, Heleganius, Guarinus, and De Chales; but the best editions are those of Dr. Barrow, 8vo. London, 1675; and Hunt, 8vo. Oxon, 1707.

There are still in existence in the National Library at Paris, two other pieces by Theodosius, one on The Cœlestial Houses, and the other on Days and Nights: a Latin translation of which was published by Peter Dasypody, A. D. 1572.

* Menelaus was a respectable mathematician and astronomer, probably of the Alexandrian school, but we have no particulars of his life or writings, except that he is said to have written six books on the chords of circular arcs, which is supposed to have been a treatise on the ancient method of constructing trigonometrical tables, but the work is lost. A Latin translation of the three books on spherical triangles was undertaken by Regiomontanus, but was first published by Maurolycus, together with the Spherics of Theodosius, and his own, (Messanæ, 1558, fol.) An edition of this work, corrected from a Hebrew manuscript, was prepared for the press by Dr. Halley, and published by Costard, the author of the History of Astronomy, in 8vo. 1758.

improved by Ptolemy, who, in the first book of his Almagest, has introduced a table of arcs and their chords, to every half degree of the semicircle; he divides the radius, and also the arc equal to one sixth of the whole circumference (whose chord is the radius) each into 60 equal parts, and estimates all other arcs by sixtieths of that arc, and their chords by sixtieths of that chord (or radius); which method he is supposed to have derived from the writings of Hipparchus, and other authors of antiquity.

No farther progress seems to have been made in the science, until some time after the revival of learning among the Arabians, namely, about the latter part of the eighth century; when the ancient method of computing by the chords of arcs was laid aside by that people, and the more convenient method of computing by the sines, substituted in its stead. This improvement has been ascribed by some to Mahomed Ebn Musa, and by others to Arzachel, a Moor, who had settled in Spain, about the year 1100: Arzachel is the first we read of who constructed a table of sines, which he employed in his numerous astronomical calculations instead of the chords, dividing the diameter into 300 equal parts, and computing the magnitude of the sines in those parts. We are indebted to the Arabs for the introduction of those axioms and theorems into the science, which are considered as the foundation of modern Trigonometry, and likewise for other improve

ments.

The sexagesimal division of the radius, according to the method of the Greeks, was still employed by the Arabians, although they had long been in possession of the Indian, or decimal scale of notation. But shortly after the diffusion of science in the west, an alteration was made by George Purbach, Professor of Mathematics at Vienna, who wrote about the middle of the 15th century; he divided the radius into 600000 equal parts and computed a table of sines in