last quotient figure is assumed greater by one than it ought strictly to be; this, as it serves only to make up for other small remainders lost, will be productive of no error of consequence in the result. 2. To find the common logarithm of the number 3. Here, by assuming A as before, the general theorem for finding the common logarithms of all numbers greater than 2, will be N=3, ... N—1=2, N—13=2×4, N-15=2x4x4, N-17= 2×4×4×4, N−1}o=2×4x4x4x4, &c. &c. whence it is plain, that the first column of divisors must be 2, 4, 4, 4, 4, &c. and the other column of divisors, in this and every other case, will be the odd numbers, 1, 3, 5, 7, &c. and proceeding as before, the work will stand thus: To which add (log. N—2=)log. 1=.000000000 The sum is the log. of 3=:477121252 In a similar manner the logarithms of the other prime numbers are obtained, and by means of them those of the composite numbers, as has been already shewn. Ans. .698970004. Ans. .S45098040. 3. To find the logarithm of 5. PART VIII. GEOMETRY. HISTORICAL INTRODUCTION. а 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 2 The name Geometry is derived from yn the earth, and ergs 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 cousistence. 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, с Thales ranks among the earliest of the Grecian philoso < 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; Augus tine 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 be 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 goon 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 |