) is most likely an abbreviation for the letters αρ; the compendia for the powers of the unknown are ΔΥ for δυναμις, the square, ΚΥ for κυβος, the cube, and so on. Diophantus shows that he solved quadratic equations by rule, like Heron. His Arithmetica, of which six books only (out of thirteen) survive, contains a certain number of problems leading to simple equations, but is mostly devoted to indeterminate or semi-determinate analysis, mainly of the second degree. The collection is extraordinarily varied, and the devices resorted to are highly ingenious. The problems solved are such as the following (fractional as well as integral solutions being admitted): ‘Given a number, to find three others such that the sum of the three, or of any pair of them, together with the given number is a square’, ‘To find four numbers such that the square of the sum plus or minus any one of the numbers is a square’, ‘To find three numbers such that the product of any two plus or minus the sum of the three is a square’. Diophantus assumes as known certain theorems about numbers which are the sums of two and three squares respectively, and other propositions in the Theory of Numbers. He also wrote a book On Polygonal Numbers of which only a fragment survives.
With Pappus and Diophantus the list of original writers on mathematics comes to an end. After them came the commentators whose names only can be mentioned here. Theon of Alexandria, the editor of Euclid, lived towards the end of the fourth century A. D. To the fifth and sixth centuries belong Proclus, Simplicius, and Eutocius, to whom we can never be grateful enough for the precious fragments which they have preserved from works now lost, and particularly the History of Geometry and the History of Astronomy by Aristotle’s pupil Eudemus.
Such is the story of Greek mathematical science. If anything could enhance the marvel of it, it would be the consideration of the shortness of the time (about 350 years) within which the Greeks, starting from the very beginning, brought geometry to the point of performing operations equivalent to the integral calculus and, in the realm of astronomy, actually anticipated Copernicus.
T. L. Heath.
NATURAL SCIENCE
Aristotle
There is a little essay of Goethe’s called, simply, Die Natur. It comes among those tracts on Natural Science in which the poet and philosopher turned his restless mind to problems of light and colour, of leaf and flower, of bony skull and kindred vertebra; and it sounds like a prose-poem, a noble paean, eulogizing the love and glorifying the study of Nature. Some twenty-five hundred years before, Anaximander had written a book with the same title, Concerning Nature, περι φυσεως: but its subject was not the same. It was a variant of the old traditional cosmogonies. It told of how in the beginning the earth was without form and void. It sought to trace all things back to the Infinite, το απειρον—to That which knows no bounds of space or time but is before all worlds, and to whose bosom again all things, all worlds, return. For Goethe Nature meant the beauty, the all but sensuous beauty of the world; for the older philosopher it was the mystery of the Creative Spirit.
Than Nature, in Goethe’s sense, no theme is more familiar to us, for whom many a poet tells the story and many a lesser poet echoes the conceit; but if there be anywhere in Greek such overt praise and worship of Nature’s beauty, I cannot call it to mind. Yet in Latin the divini gloria ruris is praised and Natura daedala rerum worshipped, as we are wont to praise and worship them, for their own sweet sakes. It is one of the ways, one of the simpler ways, in which the Roman world seems nearer to us than the Greek: and not only seems, but is so. For compared with the great early civilizations, Rome is modern and of the West; while, draw her close as we may to our hearts, Greece brings along with her a breath of the East and a whisper of remote antiquity. A Tuscan gentleman of to-day, like a Roman gentleman of yesterday, is at heart a husbandman, like Cato; he is ruris amator, like Horace; he gets him to his little farm or vineyard (O rus, quando te aspiciam!), like Atticus or the younger Pliny. As Bacon praised his garden, so does Pliny praise his farm, with its cornfields and meadowland, vineyard and woodland, orchard and pasture, bee-hives and flowers. That God made the country and man made the town was (long before Cowper) a saying of Varro’s; but in Greek I can think of no such apophthegm.