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Ulrico Hoepli, Milano). Professor Loria arranges his material in five Books, (1) on pre-Euclidean geometry, (2) on the Golden Age of Greek geometry (Euclid to Apollonius), (3) on applied mathematics, including astronomy, sphaeric, optics, &c., (4) on the Silver Age of Greek geometry, (5) on the arithmetic of the Greeks. Within the separate Books the arrangement is chronological, under the names of persons or schools. I mention these details because they raise the question whether, in a history of this kind, it is best to follow chronological order or to arrange the material according to subjects, and, if the latter, in what sense of the word 'subject' and within what limits. As Professor Loria says, his arrangement is a compromise between arrangement according to subjects and a strict adherence to chronological order, each of which plans has advantages and disadvantages of its own'.

In this book I have adopted a new arrangement, mainly according to subjects, the nature of which and the reasons for which will be made clear by an illustration. [Take the case of a famous problem which plays a great part in the history of Greek geometry, the doubling of the cube, or its equivalent, the finding of two mean proportionals in continued proportion between two given straight lines. Under a chronological arrangement this problem comes up afresh on the occasion of each new solution. Now it is obvious that, if all the recorded solutions are collected together, it is much easier to see the relations, amounting in some cases to substantial identity, between them, and to get a comprehensive view of the history of the problem. I have therefore dealt with this problem in a separate section of the chapter devoted to 'Special Problems', and I have followed the same course with the other famous problems of squaring the circle and trisecting any angle.

Similar considerations arise with regard to certain welldefined subjects such as conic sections. It would be inconvenient to interrupt the account of Menaechmus's solution of the problem of the two mean proportionals in order to

consider the way in which he may have discovered the conic sections and their fundamental properties. It seems to me much better to give the complete story of the origin and development of the geometry of the conic sections in one place, and this has been done in the chapter on conic sections associated with the name of Apollonius of Perga. Similarly a chapter has been devoted to algebra (in connexion with Diophantus) and another to trigonometry (under Hipparchus, Menelaus and Ptolemy).

At the same time the outstanding personalities of Euclid and Archimedes demand chapters to themselves. Euclid, the author of the incomparable Elements, wrote on almost all the other branches of mathematics known in his day. Archimedes's work, all original and set forth in treatises which are models of scientific exposition, perfect in form and style, was even wider in its range of subjects. The imperishable and unique monuments of the genius of these two men must be detached from their surroundings and seen as a whole if we would appreciate to the full the pre-eminent place which they occupy, and will hold for all time, in the history of science.

The arrangement which I have adopted necessitates (as does any other order of exposition) a certain amount of repetition and cross-references; but only in this way can the necessary unity be given to the whole narrative.

One other point should be mentioned. It is a defect in the existing histories that, while they state generally the contents of, and the main propositions proved in, the great treatises of Archimedes and Apollonius, they make little attempt to describe the procedure by which the results are obtained. I have therefore taken pains, in the most significant cases, to show the course of the argument in sufficient detail to enable a competent mathematician to grasp the method used and to apply it, if he will, to other similar investigations.

The work was begun in 1913, but the bulk of it was written, as a distraction, during the first three years of the

war, the hideous course of which seemed day by day to enforce the profound truth conveyed in the answer of Plato to the Delians. When they consulted him on the problem set them by the Oracle, namely that of duplicating the cube, he replied, 'It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another'.

Truly

Greece and her foundations are
Built below the tide of war,
Based on the crystalline sea
Of thought and its eternity.

T. L. H.

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