ALEXANDRIAN SCIENCE. 51 



A./ /. 



laboriously compared and sifted by Ptolemy. It is nOfeWorthy that 

 in this work Ptolemy ^ives us the first notions of the construction of 

 maps, and in it he makes use of the method of fixing the positions of 

 13Taces~"by their latitude and longitude, whichjiad been introduced by 

 Hipjtarchus. Ptojemy's geography, like his astronomy, continued to 

 be the great text-book of its subject down to the 'rise ot .modern 

 science. Nor does the list of the labours of Ptolemy close here, for 

 he left a treatise on optics, in which he discussed the subject of atmo- 

 spheric refraction, the laws of reflection from spherical mirrors, and 

 other subjects. We have also a treatise on music by Ptolemy, and 

 ancient authors mention also works on mechanics and other subjects. 



After the age of Ptolemy there were many cultivators of the mathe- 

 matical sciences at Alexandria. One of the most noted of these is 

 DIOPHANTUS (fourth century), who is^often named as the inventor of 

 algebra. He olevoted much attention to the kind of problems called 

 indeterminate, and these were in consequence known afterwards by the 

 name of the Diophantine Problems. In the fourth century there also 

 flourished at Alexandria the distinguished geometer PAPPUS. In his 

 work called " Collectiones Mathematicae " he discusses questions re- 

 lating to geometrical maxima and minima. The nature of these 

 questions may be understood by a single illustration : Let a number of 

 regular figures be given, as a square, a hexagon, an octagon, a circle, etc., 

 such that all these figures shall have their boundaries of one and- the 

 same length : the question may be asked, which of these " isoperime- 

 trical " figures has the greatest area ? The investigation of this point by 

 mathematical reasoning is not a very simple matter, but Pappus suc- 

 ceeded in solving the problem for a variety of cases. But what is most 

 curious is the application of mathematical doctrine which the Alex- 

 andrian mathematician makes to the form of the cells of the honey- 

 comb. He shows that of the regular figures only equilateral triangles, 

 squares, and hexagons would occupy a space without leaving interstices. 

 But the discussion of isoperimetrical figures having shown that the 

 hexagon contains for the same boundary the greatest area of any of these 

 three figures, the economy of material resulting from the form of the 

 bees' cells is demonstrated, and this induces Pappus to exclaim, " The 

 bees work with a kind of geometrical forethought." Contemporary with 

 Pappus was THEON OF ALEXANDRIA, whose Commentaries on Euclid 

 and on Ptolemy's Syntaxis have come down to us. 



The beautiful and hapless HYPATIA (A.D. ? 415), the daughter of 

 Theon, presents us with the rare phenomenon of a distinguished female 

 cultivator of mathematics and philosophy. She wrote Commentaries 

 on the works of Apollonius and of Diophantus, and calculated some 

 astronomical tables. Of her writings none are extant but the Com- 

 mentary on the Third Book of Ptolemy's Syntaxis, which Theon ex 

 pressly acknowledges to be hers. She lectured at Alexandria on mathe- 

 matics and the Neo-Platonic philosophy. But science and philosophy 



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