Page:Dictionary of Greek and Roman Biography and Mythology (1870) - Volume 2.djvu/80

66 are made which are not formally set down among the postulates. Things which really ought to have been pioved are souietimes passed over, and whether this is by mistake, or by intention of supposing them self-evident, cannot now be known : for Euclid never refers to previous propositions by name or number, but only by simple re-assertion without reference ; except that occasionally, and chiefly when a negative proposition is referred to, such words as "it has been demonstrated" are employed, without further specification.

Fifthly. Euclid never condescends to hint at the reason why he finds himself obliged to adopt any particular course. Be the difficulty ever so great, he removes it without mention of its exist- ence. Accordingly, in many places, the unassisted student can only see that much trouble is taken, without being able to guess why.

What, then, it may be asked, is the peculiar merit of the Elements which has caused them to retain their ground to this day.^ The answer is, that the preceding objections refer to matters which can be easily mended, without any alter- ation of the main parts of the work, and that no one has ever given so easy and natural a chain of geometrical consequences. There is a never erring truth in the results ; and, though there may be here and there a self-evident assumption used in demonstration, but not formally noted, there is never any the smallest departure from the limit- ations of construction which geometers had, from the time of Plato, imposed upon themselves. The strong inclination of editors, already mentioned, to consider Euclid as perfect, and all negligences as the work of unskilful commentators or interpo- lators, is in itself a proof of the approximate truth of the character they give the work ; to which it may be added that editors in general prefer Euclid as he stands to the alterations of other editors.

The Elements consist of thirteen books written by Euclid, and two of which it is supposed that Hypsicles is the author. The first four and the sixth are on plane geometry ; the fifth is on the theory of proportion, and applies to magnitude in general ; the seventh, eighth, and ninth, are on arithmetic ; the tenth is on the arithmetical cha- racteristics of the divisions of a straight line ; the eleventh and twelfth are on the elements of solid geometry; the thirteenth (and also the fourteenth and fifteenth) are on the regular solids, which were so much studied among the Platonists as to bear the name of Platonic, and Avhich, according to Proclus, were the objects on which the Elements were really meant to be written.

At the commencement of the first book, undor the name of definitions (Spot), are contained the assumption of such notions as the point, line, &c.. and a number of verbal explanations. Then fol- low, under the name of postulates or demands (aiT'/inaTa), all that it is thought necessary to state as assumed in geometry. There are six postulates, three of which restrict the amount of construction granted to the joining two points by a straight line, the indefinite lengthening of a terminated straight line, and the drawing of a circle with a given centre, and a given distance measured from that centre as a radius ; the other three assume the equality of all right angles, the much disputed property of two lines, which meet a third at angles less than two right angles (we mean, of course, much disputed as to its propriety as an assumption, not as to its truth), and that two straight lines cannot inclose a space. Lastly, under the name oi common notions {koipuI ei-uoiai) are given, either as com.mon to all men or to u'il sciences, such assertions as that — things equal to the same are equal to one another — the whole is greater than its part — &c. Modern editors have put the last three postulates at the end of the common notions, and applied the term aaiom (which was not used till after Euclid) to them all. The in-" tention of Euclid seems to have been, to dip^Jn- guish between that which his reader must grant, or seek another system, whatever may be his opi- nion as to the propriety of the assumption, and that which there is no question ever}-- one will grant. The modern editor merely distinguishes the assumed problem (or construction) from the assumed theorem. Now there is no such distinct tion in Euclid as that of problem and theorem ; the common term irpSraa-is, translated proposition^ includes both, and is the only one used. An im- mense preponderance of manuscripts, the testi- mony of Proclus, the Arabic translations, the summary of Boethius, place the assumptions about right angles and parallels (and most of them, that about two straight lines) among the postulates ; and this seems most reasonable, for it is certain that the first two assumptions can have no claim to rank among common notions or to be placed in the same list with " the whole is greater than its part."

Without describing minutely the contents of the first book of the Elements, we may observe that there is an arrangement of the propositions, which will enable any teacher to divide it into sections. Thus propp. 1 — 3 extend the power of construction to the drawing of a circle with any centre and ani/ radius ; 4 — 8 are the basis of the theory of equal triangles ; 9 — 12 increase the power of construction ; 13 — 15 are solely on rela- tions of angles; 16 — 21 examine the relations of parts of one triangle ; 22 — 23 are additional con- structions ; 23 — 26 augment the doctrine of equal triangles ; 27 — 31 contain the theory of parallels; 32 stands alone, and gives the relation between the angles of a triangle ; 33 — 34 give the first properties of a parallelogram ; 35 — 41 consider parallelograms and triangles of equal areas, but different forms; 42 — 46 apply what precedes to augmenting power of construction ; 47 — 48 give the celebrated property of a right angled triangle and its converse. The other books are all capable of a similar species of subdivision.

The second book shews those properties of the rectangles contained by the parts of divided straight lines, which are so closely connected with the common arithmetical operations of multipli- cation and division, that a student or a teacher who is not fully alive to the existence and diffi- culty of incommensurables is apt to think that common arithmetic would be as rigorous as geo- metry. Euclid knew better.

The third book is devoted to the consideration of the properties of the circle, and is much cramped in several places by the imperfect idea already al- luded to, which Euclid took of an angle. There are some places in which he clearly drew upon experimental knowledge of the form of a circle,