iv 



THE PROGRESS OF 



which are repeated every saros in the same 

 ordei. 



The Chaldeans appear to have divided the day 

 into twelve hours, and to have constructed sun- 

 dials for pointing out the hour. The sun-dial of 

 Ahaz is mentioned in the Old Testament, on the 

 occasion of the recovery of Hezekiah ; but no- 

 thing is said about its construction. Undoubt- 

 edly, however, such sun-dials would require a 

 in knowledge of gnomonics, which therefore 

 the Chaldeans must have possessed. 



That the l-'ijyptians hail made somo progress 

 in mathematics admits of no doubt, as the Greeks 

 inform us that they derived their first knowledge 

 of that branch of science from the Egyptian 

 priests. Hut that the mathematical knowledge 

 of that people could not have been very exten- 

 sive, is evident from the ecstasy into which 

 Pythagoras was thrown, when he discovered that 

 the square of the hypothenuse of a right angled 

 triangle is equal to the square of the two sides. 

 For ignorance of this very elementary, but im- 

 portant proposition, necessarily implies very 

 little knowledge even of the most elementary 

 parts of mathematics. 



It was in Greece that pure mathematics first 

 made decided progress. The works of three Greek 

 mathematicians still remain, from which we 

 have obtained information of all or almost all the 

 mathematical knowledge attained by the Greeks. 

 These are Euclid, Apollonius, and Archimedes. 



Euclid lived in Alexandria during the reign 

 of the first Ptolemy. Nothing whatever is known 

 respecting the place of his nativity ; though it i 

 certain that he lived in Greece, and that he died 

 in Egypt, after the foundation of the celebrated 

 Alexandrian school. He collected all the ele- 

 mentary facts known in mathematics before his 

 time, and arranged them in such an admirable 

 order beginning with a few simple axioms, and 

 deducing from them his demonstrations, every 

 subsequent demonstration depending on, and 

 rigidly deduced from those that immediately 

 precede it that no subsequent writer has been 

 able to produce any thing superior or even 

 equal His Elements still continue to be taughi 

 in our schools, and could not be dispensed with 

 unless we were to give up somewhat of tha 

 rigour which has been always so much admirei 

 in the Greek geometricians. Perhaps, however 

 we carry this admiration a little too far. Th< 

 geometrical axioms might be somewhat enlarged 

 without drawing too much upon the faith of be 

 ginners. And were that method followed, con 

 siderable progress might be made in mathema 

 tics without encountering some of those difficul 

 demonstrations that are apt to damp the ardou 

 of beginners. 



The Elements of Euclid consist of thirteen 

 tooks. In the first four he treats of the pro- 

 terties of lines, parallel lines, angles, triangles, 

 mil circles. The fifth and sixth treat of propor- 

 ion or ratios. The seventh, eighth, ninth, and 

 fiith treat of numbers. The eleventh and 

 twelfth treat of solids; and the thirteenth of 

 solids; also of certain preliminary propositions 

 about cutting lines in extreme and mean ratio. 

 It is the first four books of Euclid chiefly that 

 ire studied by modern geometricians. The rest 

 lave been, in a great measure, superseded by 

 more modern improvements. 



Apollonius was born at Perga in Pamphylia, 

 about the middle of the second century before 

 the Christian era. Like Euclid, he repaired to 

 Alexandria, and acquired his mathematical 

 knowledge from the successors of that geome- 

 trician. The writings of Apollonius were 

 numerous and profound; but it is upon his 

 Treatise on the Conic Sectiom, in eight books, 

 that his celebrity as a mathematician chiefly 

 depends. 



The Conic Sections, which, after the circle, 

 are the most important of all curves, were dis- 

 covered by the mathematicians of the Platonic 

 school ; though who the discoverer was is not 

 known. A considerable number of the pro- 

 perties of these curves were gradually developed 

 by the Greek geometricians. And the first four 

 books of Apollonius are a collection of every 

 thing known respecting these curves before his 

 time. The last four books contain his own dis- 

 coveries. In the fifth book he treats of the 

 greatest and smallest lines which can be drawn 

 from each point of their circumference, and 

 many other intricate questions, which required 

 the greatest sagacity and the most unremitting 

 attention to investigate. The sixth book is not 

 very important nor difficult ; but the seventh 

 contains many very important problems, and 

 points out the singular analogy that exists be- 

 tween the properties of the various conic sections. 

 The eighth book has not come down to us. The 

 fifth, sixth, and seventh books were discovered 

 by Borelli, in Arabic, in the library of the grand 

 duke of Tuscany. He got them translated, and 

 published his translation, with notes and illustra- 

 tions, in the year 1661. I)r Halley published an 

 edition of Apollonius in 1710, and has supplied 

 the eighth book from the account given by Pap- 

 pus of the nature of its contents. 



Archimedes was, beyond dispute, the greatest 

 mathematician that antiquity produced. He was 

 born in Sicily about the year 287 before the 

 Christian era, and is said to have been a relation 

 of Hiero, king of Syracuse. So ardent a cul- 

 tivator was ho of the mathematics, that he was 



