The first question admits of an infinite number of answers, and the second of only one; or, in the language of Euclid, if the area, ratio of aides, and an angle of,a parallelogram be given, the aides themselves are given. The same process by which it may be shown that they are given serves to find them ; so that the Data of Euclid may be looked upon as a collection of geometrical problems, in which the attention of the reader is directed more to the question of the sufficiency of the hypothesis to produce one result, and one only, than to the method of obtaining the result.
A preface to was written by one Marinus, the disciple and successor of Proclui, explaining at tedious length the distinction of ' given' and 'not given.' 10. /voisse7a, the 'Elements' (of Geometry). For a long time writers hardly considered it necessary to state whose ' Elements' they referred to, since a certain book of the elements always signified that book of Euclid ; and it was customary in England to call each book an element ; thus in Billingsley's old translation the sixth hook is called the sixth element.
The reason why the ' Elements' have maintained their grouni is not their extreme precision in the statement of what they demand ; for it frequently happens that a result is appealed to ae self-evident which is not to be found hi the expressed axioms. Neither does their fame arise from their never assuming what might be proved ; for in the very definitions we find it asserted that the diameter of a circle bieects the figure, which might be readily proved from the axioms. Neither is it the complete freedom from redundancy, nor the perfec tion of the arrangement; for book L, prop. 4, which le very much out of place, considering that it is never wautod in the first book, is, in point of fact, proved again, though not expressed, In prop. 19. Neither is it the manner in which our ideas of magnitude are rendered com plete, as well as definite : for instance, book iii. prop. 20 Is incomplete without Euclid's definition and use term ' angle ;' nor with that term, as used by him, can the 21st proposition of that book be fully demonstrated without the help of the subsequent 22nd. In fact, the Elementa• abouod in defects, which, if we may so speak, are clearly seen by the light of their excellences ; the high standard of accuracy which they ineulesto in general, the positive and explicit statement which they make upon all real and important assumptions, the natural character of the arrangement, the complete and perfect absence of false conclusion or fallacious reasoning. and the judicious choice of the demonstrations, considered with reference to the wants of the beginner, are the causes of the universal celebrity which this book has enjoyed.
We shall now describe the contents of the ' Elemeuts.' There are thirteen books certainly written by Euclid, and two more (the fourteenth and fifteenth) which are supposed to have been added by Ilypeicles of Alexandria (about A. D. 170).
Book I. lays down the definitions and postulates required in the establishment of plane geometry, a few definitions being prefixed also to books iv., and vi. It then treats of such properties of straight lines and triangles as do not require any particular consideration of the properties of the circle nor of proportion. It contains the cele brated proposition of Pythagoras.
From this book it appears that Euclid lays down, as all the iostru mental aid permitted in geometry, the description of a right line of indefinite length, the indefinite continuation of such right line, and the description of • circle with a given centre, the circumference of which is to pass through a given point It is usual to say, then, that the role and compasses are the instruments of Euclid's geometry, which is not altogether correct, unless it be remembered that with neither ruler nor compasses is • straight line allowed to be transferred, of a given length, from one part of apace to another. It is a plain ruler, whore soda are not allowed to be touched, and compasses which close the moment they are taken off the paper, of which the Greek geometry permits the use. It is altogether uncertain by whom these restrictive postulates were introduced, but it must have been before the time of Plato, who was contemporary with (if he did not come after) the introduction of those problems whose difficulty depends upon the restrictions. We may here observe that in actual construe• tion the ruler might have been dispensed with. It was reserved fin an Italian abbe, at the end of the 18th century, when all who studied geometry had for two thousand years admired the smallness of the bares on which its conclusions are built, to inquire whether, small at they were, less would not have been sufficient In Ma:whereon ' Geometria del Compasses; published at Pavia in 1797, it is 'howl] that all the fundamental constructions of geometry can be made with out the necessity of determining any point by the intersections of atenight lines ; that is, by using only those of circles. This singular and very original work was translated into French, and published al Paris in 17113 and 1828.