1.  A LARGE book might be composed, were we but cursorily to mention al the important discoveries in geometry, mathematics and philosophy, for which we are indebted to the ancients.  Wherefore, not to swell this volume, we shall just point at the principle of this without insisting at length, because it is generally acknowledged that they owe their origin to those philosophers of antiquity.

2.  All the learned agree, that Thales was the first we know who predicted eclipses; pointed out the advantages that must all from a due observation of the little bear, or polar star; taught that the earth was round, and the ecliptic in an oblique position.  He had no less service to geometry than astronomy.  He instructed in that science the Egyptions themselves, to who he went to be taught.  He showed them how to measure the pyramids by the length and distances, by the proportion of the sides of a triangle.  He demonstrated the various properties of the circle; particularly that whereby it appears that all triangles which have the diameter for their base, the subtending angle of which touches the circumference are in that point of contact right angled.  He discovered respecting the isosceles triangle, that the angles at its vase were equal; and was the first who found out, that in right lines cutting one another, the opposite angles are equal,.  In short, he taught a great may other valuable truths, too long to be narrated.  We owe to Anaximander, the successor of Thales, the invention fo the armillary sphere, and of sun dials; he was likewise the first who drew a geographical map.

3.   Pythagoras has already afforded to us many instances of his profound knowledge in all the sciences.  There are few philosophers, even among the ancients, who had so much sagacity and depth of genius,  He was the first who gave sure and fundamental precepts, with respect to music, which he fixed upon by a reach of discernment that was extraordinary.  Struck by the difference of sounds which issued from the hammers of a forge, but came into unison at the fourth, and fifth, and eighth percussions; he concluded that this must proceed, from the difference of weight of the hammers; he had them weighed, and found that he had conjectured right.  Upon this he wound up some musical strings, in number equal to the hammers, and of a length proportional to their weight; and found, that at the same intervals, they corresponded with the hammers in sound.  It was upon the same principle that he devised the monochord; an instrument consisting of one string, yet capable of easily determining the various relations of sound. He also made many fine discoveries in geometry, among others, that-property of a right angled triangle, that the square of the hypothenuse, or side subtending the right angle, is equal to the squares of the two other sides. And he gave the first sketch of the doctrine of isoperimeters, in demonstrating, that of all plane figures, the circle is the largest; and of all solids, the sphere.

4.   Plato likewise applied himself to the study of mathematics; and we owe to him many fine discoveries in that science. He it was, who first introduced the analytic method, or that geometric analysis which enables us to find the truth we are in quest of, out of the proposition itself which we want to resolve. He it was, who at length solved the famous problem, respecting the duplication of the cube, on account of which so much honour is paid, by all the philosophers of his school, to Euxodes, Archydes and Menechinus; To him also, is ascribed the solution of the problem concerning the tri section of an angle; and the discovery of the conic sections. Pappus. hath given us the summary of a great many analytic works. In the preface to his seventh book, we meet with this principle of Guldinus, “that whatever figure arises from the circumvolution of another, is produced by the revolution of the latter about its centre of gravity.”

5.   Geometry is indebted to Hipparchus for the first elements of plain and spherical trigonometry; and to Diophantes, who lived three hundred and sixty years before Jesus Christ, we owe the invention of algebra. That the ancients laid the first foundation of algebra, is a thing out of doubt, as shown by the celebrated Wallis, in his history of that science. He makes no question but algebra was known to the ancients, and that they thence drew those long and difficult demonstrations which we meet with in their works. He supports his opinion by the testimonies of Schoten, Oughtred and Barrow; and makes mention of a manuscript in the Savillian library, which treats of this science, and bears the name of Appollonius. But he thinks the ancients carefully concealed a method, which furnished them with so many beautiful and difficult demonstrations; and that they chose rather to prove their propositions by reasonings ad absurdum, than to hazard the -discovery of that method, which brought them more directly to the result of what they demonstrated. One to whom algebra is much indebted, Leibnits, forms the same judgment. Speaking of the higher operations of it, he says, “in perusing the arithmetic of Diophantes, and the geometrical books of Apollonius and Pappus, we cannot doubt, but the ancients had some knowledge of it. Vietus extended it still further, in expressing by those general characters, not only unknown numbers and proportions, but such as are known ; doing that by figures, which Euclid does by reasoning. And Descartes hath extended it to geometry, in marking by equations the proportions of lines. Yet, even since the discovery of our modern algebra, Mr. Bouillaud, whom I was acquainted with at who was without all doubt, an excellent geometrician, never reflect, but with astonishment, on the demonstrations of Archimedes concerning the properties of the spiral line, and could not how that great man hit upon the applying the tangent of- that line to the commensuration of the circulation of the circle.” Nunes is of the same opinion with the former ; and in his history of Algebra, regrets that the ancients concealed from US, a method which they-themselves used; and says, “that we are not to think that the greater part of the propositions of Euclid and Archimedes, were founded by those great men in that way of reasoning, in which they have thought proper to transmit them to us.”

6.  This method of theirs, which resembled our algebra, sometimes however, discovers itself in their researches. We meet with traces of it sufficiently strong in the thirteenth book of Euclid; especially, if we make use of the Greek text, or the old Latin translation. And although Wallis imagines, that they may belong to some other scholiasts; yet the antiquity of the science itself will still be the same. Some indeed make it mount much higher, who, led by the authority of some able mathematicians among the ancients, assign the first invention of it to Plato. Whoever desires to enter into a more exact examination of this, will find in Wallis a guide and monitor, whose authority may be acquiesced in, he having set this matter in the-clearest light, as well as made the first and noblest efforts in our time, to raise algebra to the state of perfection which it bath now attained. Now, according to this able geometrician, the method of investigating infinite serieses took its rise from his arithmetic of infinites, published in 1656; and he himself acknowledges, that both of them are founded on the method of exhaustions, used by the ancients. He farther says, “that the method of indivisibles introduced ;by cavallieri, is no other than an abridgment of that of exhaustions, though somewhat more obscure.” He observes, that the lines and surfaces, whose proportion and contents are inquired into, and ascertained by Cavallieri, differ in nothing from the inscribed and circumscribed triangles, whose approaches Archimedes brought so near, that the difference of space enclosed between them, and that which they approached, and about which they were drawn, to wit, the -contents of the circle, might become less than any assignable quantity: and this he proves afterward, by an analytical exposition of both. I may however remark, that from the time of Diophantes, algebra made but small progress, till that of Vietus, who restored and perfected it, and was the first who marked the known quantities by the letters of the alphabet.

7.  Besides the discoveries made in astronomy by the ancients, which we have been reading, there are a great many others, which I cannot bring into view, in that full manner they deserve. Yet I cannot omit mentioning here one important observation of Aristarchus. was the first who suggested a method of measuring the distance of the sun from the earth, by means of the half section of the moon’s disk, or that basis of it wherein it appears to us when it is in its quadratures.

            8.   Hipparchus was the first who calculated tables of the motion of the sun and moon, and composed a catalogue of the fixed stars. He was also the first, who, from the observation of eclipses, determined the longitude of places upon earth : but what above all does immortal honour to his genius is, that he laid the first foundations for the discovery of the precession of the equinoxes. Mr. Bayle reprebends Rohault as laying under a mistake, when he says that, “Hipparchus knew nothing of the peculiar motion of the fixed stars from west to east, which is the cause of their varying the longitude.”— Yea, and Timoeus Locrensis, who lived before Plato, taught this very astronomical truth in clear terms.’

Chapter 20