PART VIII.

GEOMETRY.

HISTORICAL INTRODUCTION.

GEOMETRY is the science of magnitude, or local extension; it teaches and demonstrates the properties of lines, surfaces, solids, ratios, and proportions, in a general manner, and with the most unexceptionable strictness and precision. Geometry, or measuring, must have been practised as an art at the commencement of society, or shortly after, when men began to build, and to mark out the limits of their respective territories. That this art had reached a considerable degree of perfection at the time of the general deluge, can hardly be doubted from that stupendous monument of human folly, the Tower of Babel, which was begun about 115 years after that period: Herodotus informs us, that this vast building had a square base, each side of which was a furlong in length; Strabo affirms that its height was likewise a furlong; and Glycas says, that the constant labour of forty years was consumed in erecting this unfinished and useless fabric. The Pyramids, Obelisks, Temples, and other public edifices with which Egypt abounded, existed prior to any authentic date of profane history: many of these had been in ruins probably

a The name Geometry is derived from yn the earth, and wires to measure. The invention of measuring is ascribed to the Egyptians by Herodotus, Diodorus, Strabo, and Proclus; to Mercury by others among the ancients; and to the Hebrews by Josephus.

for ages before the earliest historians lived, who speak of their magnificence as surpassing that of the most splendid structures in Greece. Can it be supposed possible, that buildings, whose magnificent remains alone were sufficient to excite the wonder and admiration of a learned and polished nation like the Greeks, could have been raised without the assistance of Geometry?

The priests of Memphis informed Herodotus, that their king Sesostris divided the lands bordering on the Nile among his subjects, requiring that the possessor should pay an annual tribute proportionate to the dimensions of the land he occupied; and if the overflowing of that river occasioned any diminution, the king, on being applied to, caused the land to be measured, and claimed tribute in proportion only to what remained. "I believe," adds Herodotus, "that here Geometry took its birth, and hence it was transmitted to the Greeks." On the strength of this conjecture we frequently hear it affirmed, that "Geometry derived its origin from the annual inundation of the Nile;" but it is plain that this as

b Several instances of this may be given. The tomb of Osymandyas, one of their kings, is said to have been uncommonly magnificent; it was surrounded by a circle of gold, 365 cubits in circumference, divided into as many equal parts, which shewed the rising and setting of the sun for every day in the year: this circle was carried away by Cambyses, king of Persia, when he conquered Egypt, A. C. 525. Goguet Orig. des Lois, &c. T. 2. liv. 3. Rollin's Anc. Hist. vol. I. p. 3. The famous Labyrinth contained 12 palaces surrounded by 1500 rooms, adorned with innumerable ornaments and statues of the finest marble, and most exquisite workmanship; there were besides, 1500 subterraneous apartments, which Herodotus (who surveyed this noble and beautiful structure) was not permitted to see, because the sepulchres of their kings were there, and likewise the sacred crocodiles and other animals, which a nation so wise in other respects worshipped as gods: "Who" (says the learned and pious Rollin) "can speak this without confusion, and without deploring the blindness of man!" The magnificent city of Thebes, with its numerous and splendid palaces and other public edifices, which was ruined by Cambyses, is the last instance to be mentioned, although many more might be added. It extended above 23 miles, had an hundred gates, and could send out at every gate 20,000 fighting men, and 200 chariots.

sertion deserves little credit; for as a science, Geometry never existed in Egypt before the time of Alexander, and as an art it must have been known there (as we have shewn above) long before the age of Sesostris; for according to the very probable conclusions of our most accurate and best informed chronologers, Sesostris was the Egyptian king, who invaded Jerusalem, A. C. 971; on which occasion he is mentioned in 1 Kings, ch. xiv. v. 25, under the name of Shishak: now we have direct proofs, on the most unquestionable authority, that measuring was understood by the Jews who came from Egypt, many centuries earlier than that date; see Genesis, ch. vi. v. 15, 16. Exodus, ch. xxv. xxvi. xxvii. and various other parts of the Mosaic history.

Not to take up the reader's time with conjectures about the origin of Geometry, which at best must be vague and uncertain, we hasten to inform him, that the Greeks, to whose taste and industry almost every science stands indebted, were the first people who collected the scattered principles and practices of Geometry, which they found in Egypt and other eastern countries, and moulded them into a form and consistence. Until it passed through their masterly hands, Geometry could not by any accommodation of language be properly termed a science; but by their consummate skill and indefatigable labours, a few scanty and detached principles and rules, heretofore chiefly applied to the measuring of land, (as the name Geometry imports,) at length grew into and became the most complete and elegant science in the world. We adore that benign Providence, who has repeatedly condescended to make even wicked and idolatrous nations useful instruments for promoting the execution of his merciful designs to man.

c Thales, the father of the Greek philosophy, and the first of the seven wise men of Greece, was born at Miletum, A. C. 640; after acquiring the best learn

phers, who travelled into foreign countries in quest of that knowledge which their own could not supply, A. C. 640. He became not only an able geometer, but was likewise very skilful in every branch of Mathematics and Physics, as these sciences then stood. We are unacquainted with the particulars of his acquirements and discoveries in Geometry, but he is mentioned as being the first who measured the height of the pyramids at Memphis, by means of their shadows, and who applied the circumference of a circle to the measuring of angles.

Pythagoras" was another eminent Grecian philosopher, who

ing his own country afforded, he travelled in the East, and returned with a mind enriched with the knowledge of Geometry, Astronomy, Natural Philosophy, &c. which he improved by his own skill and application. He divided the celestial sphere into five zones; he observed the apparent diameter of the sun, making it half a degree; he understood the cause and course of eclipses, calculated them with accuracy, and divided the year into 365 days. He disliked monarchy, because he considered it as little better than tyranny, to every species of which he was an avowed enemy. One evening as he walked out to contemplate the stars, he had the misfortune to fall into a ditch, on which an old woman, who saw him, exclaimed, "How can you possibly know what is doing in the heavens, when you cannot see what is even at your feet!" He died at the Olympic Games, at the age of upwards of 90 years. Thales was the founder of the Ionian sect, and had for his scholars some of the most eminent philosophers of antiquity, among whom are mentioned Anaximander, Anaximenes, and Pythagoras. It is uncertain whether he left any writings; Augustine mentions some books on Natural Philosophy ascribed to him; Simplicius, some on Nautic Astrology; Laërtius, two treatises on the Tropics and Equinoxes; and Suidas, a work on Meteors, written in verse.

d Pythagoras, a celebrated philosopher of Samos. He was early instructed in music, poetry, astronomy, and gymnastic exercise, with whatever else might tend to enlighten his mind, and invigorate his body. At the age of eighteen he resolved to travel for that instruction, which the ablest philosophers of Samos were incompetent to supply: he spent 25 years in Egypt, where having ingratiated himself with the priests, he became acquainted with all the learning of that country; having travelled through Chaldea, and visited Babylon, he returned, passing through Crete, Sparta, and Peloponnesus, from whence he crossed over into Italy, and finally fixed his residence at Crotona. Here he opened a school, which, by the fame of his mental and personal accomplishments, was soon crowded with pupils, many of whom came from distant parts of Greece and Italy. His scholars, who were called the Italian sect, were formed by

was endowed with an equal thirst for useful knowledge, and employed the same means to gratify it, A. C. 590. The 32nd and 47th propositions of the first book of Euclid's Elements are ascribed to him; from the latter of which he was led to determine, that the diagonal of a square is incommensurable to its side: every person moderately acquainted with Geometry will acknowledge, that the useful purposes to which these important propositions may be applied are innumerable. About this time, or shortly after, the following celebrated

the rules obtained from the Egyptian priests; among other austerities, he enjoined them a five years silence, during which they were only to hear; after this they were allowed to propose doubts, ask questions, &c. in which they were permitted to say, not a little in many words, but much in as few words as possible. Query. Might not the prattling, self-sufficient young gentlemen in some of our academies, be admirably benefited by an institution of this kind?

Besides the propositions mentioned above, Pythagoras was the author of the following, viz. only three rectilineal figures can fill up the space about a point; namely, the equilateral triangle, the square, and the hexagon. He invented the multiplication table; the obliquity of the ecliptic was first discovered by him ; he called the world norμos, and asserted that it was made in musical proportion; the sun he called the fiery globe of unity, and maintained that the seven planets move round him in an harmonious motion at distances corresponding to the musical divisions or intervals of the monochord: he taught the true solar system, which had been asserted by Philolaus of Crotona, but being forgotten and lost during many ages after, was at length revived by Copernicus, and demonstrated by the illustrious Newton.

The modesty of Pythagoras was not less conspicuous than his attainments; on being addressed at a public assembly with the splendid appellation of ropes, wise man, he disclaimed the title, and requested that they would rather call him piλropos, a lover of wisdom; a circumstance which first gave rise to the terms philosophy and philosopher.

Some authors affirm, that Pythagoras offered 100 oxen as a sacrifice to Apollo, in gratitude for the discovery which that god enabled him to make of the 47th proposition of the first book of Euclid; this is extremely improbable, as he was a firm believer in the doctrine of the transmigration of souls, which forbade taking away the life of any animal: nor is it much more credible that he sub. stituted little oxen made of flour, clay, or wax ; no, this would doubtless have been considered as an intolerable affront, which the meanest heathen god in the catalogue would disdain to put up with. The whole story is perhaps nothing better than a fiction, an ingenious sample of ancient priest-craft.