and made tacit assumptions of a kind which are rarely met with in his writings.
The fourth book treats of regular figures. Eu- clid's original postulates of construction give him, by this time, the power of drawing them of 3, 4, 5, and 15 sides, or of double, quadruple, &c., any of these numbers, as 6, 12, 24, &c., 8, 16, &c. &c.
The fifth book is on the theory of proportion. It refers to all kinds of magnitude, and is wholly independent of those which precede. The exist- ence of incommensurable quantities obliges him to introduce a definition of proportion which seems at first not only difficult, but uncouth and inele- gant ; those who have examined other definitions know that all which are not defective are but various readings of that of Euclid. The reasons for this difficult definition are not alluded to, ac- cording to his custom ; few students therefore un- derstand the fifth book at first, and many teachers decidedly object to make it a part of the course. A distinction should be drawn between Euclid's definition and his manner of applying it. Every one who understands it must see that it is an application of arithmetic, and that the defective and unwieldy forms of arithmetical expression which never were banished from Greek science, need not be the necessary accompaniments of the modern use of the fifth book. For ourselves, we are satisfied that the only rigorous road to propor- tion is either through the fifth book, or else through something much more difficult than the fifth book need be.
The sixth book applies the theory of propor- tion, and adds to the first four books the proposi- tions which, for want of it, they could not contain. It discusses the theory of figures of the same form, technically called similar. To give an idea of the advance which it makes, we may state that the first book has for its highest point of constructive power the formation of a rectangle upon a given base, equal to a given rectilinear figure ; that the second book enables us to turn this rectangle into a square ; but the sixth book empowers us to make a figure of any given rectilinear shape equal to a rectilinear figure of given size, or briefly, to construct a figure of the form of one given figure, and of the size of another. It also supplies the geometrical form of the solution of a quadratic equation.
The seventh, eighth, and ninth books cannot have their subjects usefully separated. They treat of arithmetic, that is, of the fundamental properties of numbers, on which the rules of arithmetic must be founded. But Euclid goes further than is ne- cessary merely to constnict a system of computa- tion, about which the Greeks had little anxiety. He is able to succeed in shewing that numbers which are prime to one another are the least in their ratio, to prove that the number of primes is infinite, and to point out the rule for constructing what are called perfect numbers. When the mo- dern systems began to prevail, these books of Eu- clid were abandoned to the antiquary : our elemen- tary books of arithmetic, which till lately were all, and now are mostly, systems of mechanical rules, tell us what would have become of geometry if the earlier books had shared the same fate.
The tenth book is the development of all the power of the preceding ones, geometrical and arith- metical. It is one of the most curious of the Greek speculations : the reader will find a synoptical ac- count of it in the Feiiny Cyclopaedia, article, " Ir- rational Quantities." Euclid has evidently in his mind the intention of classifying incommensurable quantities : perhaps the circumference of the circle, which we know had been an object of inquiry, was suspected of being incommensurable with its diameter ; and hopes were perhaps entertained that a searching attempt to arrange the incommen- surables which ordinary geometry presents might enable the geometer to say finally to which of them, if any, the circle belongs. However this may be, Euclid investigates, by isolated methods, and in a manner which, unless he had a concealed algebra, is more astonishing to us than anything in the Elements, every possible variety of lines which can be represented by / {/ a+^^h), a and b repre- senting two commensurable lines. He divides lines which can be represented by this formula into 25 species, and he succeeds in detecting every possible species. He shews that every individual of every species is incommensurable with all the individuals of every other species ; and also that no line of anj' species can belong to that species in two different ways, or for two different sets of values of a and I. He shews how to form other classes of incommen- surables, in number how many soever, no one of which can contain an individual line which is com- mensurable with an individual of any other class ; and he demonstrates the incommensurability of a square and its diagonal. This book has a com- pleteness which none of the others (not even the fifth) can boast of: and we could almost suspect that Euclid, having arranged his materials in his own mind, and having completely elaborated the tenth book, wrote the preceding books after it, and did not live to revise them thoroughly.
The eleventh and twelfth books contain the elements of solid geometry, as to prisms, pyramids, &c. The duplicate ratio of the diameters is shewn to be that of two circles, the triplicate ratio that of two spheres. Instances occur of the method of exhaustions^ as it has been called, which in the hands of Archimedes became an instrument of dis- covery, producing results which are now usually referred to the differential calculus : while in those of Euclid it was only the mode of proving proposi- tions which must have been seen and beHeved be- fore they were proved. The method of these books is clear and elegant, with some striking imperfec- tions, which have caused many to abandon them, even among those who allow no substitute for the first six books. The thirteenth, fourteenth, and fifteenth books are on the five regular solids : and even had they all been written by Euclid (the last two are attributed to Hypsicles), they would but ill bear out the assertion of Proclus, that the regu- lar solids were the objects with a view to which the Elements were written : unless indeed we are to suppose that Euclid died before he could com- plete his intended structure. Proclus was an en- thusiastic Platonist : Euclid was of that school ; and the former accordingly attributes to the latter a particular regard for what were sometimes called the Platonic bodies. But we think that the author himself of the Elements could hardly have considered them as a mere introduction to a favourite specula- tion : if he were so blind, we have every reason to suppose that his own contemporaries could have set him right. From various indications, it can be col- lected that the fame of the Elements was almost coeval with their publication ; and by the time of
