373 



GEOMETRY. 



GEOMETRY. 



374 



how to measure an unknown height by the well-known law of reflected 

 light. In the second an imperfect theory of convex and concave 

 mirrors is given. 



7. Qaivoiieva, on Astronomical Appearances, mentioned by Pappus, 

 and Philoponus (cited by Gregory). It contains a geometrical doctrine 

 of the sphere, and though probably much corrupted by time, is un- 

 doubtedly Euclid's. 



8. KoraToyui? Kavivos and eitraywyh affioviK^, the Division of the Scale 

 and Introduction to Harmony. Proclus mentions that Euclid wrote 

 on harmony, but the first of these treatises is a distinct geometrical 

 refutation of the principles laid down in the second, which renders it 

 unlikely that Euclid should have written both. The second treatise 

 is Aristoxenian, while the first proceeds on principles of which 

 Gregory states he never found a vestige in any other writer who was 

 reputed anterior to Ptolemy (to whom he attributes it). The second 

 treatise is not geometrical, but is purely a description of the system 

 mentioned, and as this treatise is not alluded to by Ptolemy nor by 

 any previous writer on the subject, it is very probable that Euclid did 

 not write it. 



9. AcSo/ifVa, a book of Data. This is the most valuable specimen 

 which we have left of the rudiments of the geometrical analysis of the 

 Greeks. Before a result can be found, it should be known whether the 

 given hypotheses are sufficient to determine it. The application of 

 algebra settles both points ; that is, ascertains whether one or more 

 definite results can be determined, and determines them. But in 

 geometry it is possible to propose a question which is really indeter- 

 minate, and in a determinate form, while at the same time the methods 

 of geometry which give one answer may not give the means of ascer- 

 taining whether the answer thus obtained is the only one. Thus the 

 two following questions seem equally to require one specific answer, 

 to one not versed in geometry : 



Given, the area of a parallelogram, and the ratio of its sides; 

 required, the lengths of those sides : and 



Given, the area of a parallelogram, the ratio of its sides and one of 

 its angles ; required, the lengths of the sides. 



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 

 sides, and an angle of a parallelogram be given, the sides themselves 

 are given. The same process by which it may be shown that they are 

 given serves to find them ; HO 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 this book was written by one Marinus, the disciple and 

 successor of Proclus. explaining at tedious length the distinction of 

 " given " and " not given." 



1 0. Sroixcia, 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'a old translation the sixth book is called 

 the tixth element. 



The reason why the Elements have maintained their ground is not 

 then- extreme precision in the statement of what they demand 

 [AXIOM] ; for it frequently happens that a result is appealed to as 

 self evident, which is not to be found in 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 bisects the figure, which might be readily proved 

 from the axioms. Neither is it the complete freedom from redundancy, 

 nor the perfection of the arrangement ; for book i. prop. 6, which is very 

 much out of place, considering that it is never wanted 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 

 complete,as well as definite : for instance, book iii. prop. 20, is incomplete 

 with Euclid's definition and use of the term angle ; nor with that term 

 as used by him can the 21st proposition of that book be fully demon- 

 strated withi.ut the help of the subsequent 22nd. In fact, the Elements 

 abound in defects, which, if we may so apeak, are clearly seen by the 

 light of then- excellencies : the high standard of accuracy which they 

 inculcate 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, 

 consiilered with reference to the wants of the beginner, are the causes 

 of ^he universal celebrity which this book has enjoyed. We shall, in 

 the article MATHEMATICS, give our reasons for advocating the conti- 

 nuance of Euclid as a book of instruction, and shall now describe the 

 contents of the Element*. 



There are thirteen books certainly written by Euclid, and two 



more (the fourteenth and fifteenth) which are supposed to have been 



>iy HyimicIeB of Alexandria, of whom we do not doubt that 



he lived in the sixth century, though he in commonly placed in the 



Moond, 



Book i. lays down the definitions and postulates required in the 

 establishment of plane geometry, a few definitions being prefixed also 

 to ii., iii., 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 celebrated 

 proposition of Pythagoras. 



From this book it appears that Euclid lays down, as all the instru- 

 mental aid permitted in geometry, the description of a right line of 

 indefinite length, the indefinite continuation of such a right line, and 

 the description of a circle with a given centre, the circumference of 

 which is to pass through a given point. It is usual to say, then, that 

 the ruler 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 a straight line allowed to be transferred, 

 of a given length, from one part of space to another. It is a plain 

 ruler, whose ends 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 construction the 

 ruler might have been dispensed with. It was reserved for an Italian 

 abbe 1 , at the end of the 18th century, when all who studied geometry 

 had, for two thousand years, admired the smalluess of the bases on 

 which its conclusions are built, to inquire whether, small as they were, 

 less would not have been sufficient. In Mascheroni's ' Geometria del 

 Compassa,' published at Pavia in 1797, it is shown that all the 

 fundamental constructions of geometry can be made without the 

 necessity of determining any point by the intersections of straight 

 lines ; that is, by using only those of circles. Tliis singular and very 

 original work was translated into French, and published at Paris in 

 1798 and 1828. It may be added that Benedetti, in the 16th century, 

 and others after him, had shown that, granting the straight line, OH/I/ 

 one circle is absolutely necessary : that is, only one opening of the 

 compasses. 



On subjects particularly connected with the first book, see AXIOM, 

 POSTULATE, PABALLELS, HYPOTHENUSE. 



Book ii. treats of the squares and rectangles described upon the parts 

 into which a line is divided. It opens the way for the application of 

 geometry to arithmetic, and ends by showing how to make a square 

 equal to any rectilinear figure. It also points out what modification 

 the proposition of Pythagoras undergoes in ,the case of a triangle not 

 right-angled. [RECTANULE.] 



Book iii. treats of the circle, establishing such properties as can be 

 deduced by means of the preceding books. 



Book iv. treats of such rei^ular figures as can readily be described by 

 means of the circle only, including the pentagon, hexagon, and quiude- 

 cagon. It is of no use in what immediately follows. 



Book v. treats of proportion generally, that is, with regard to 

 magnitude in general. Whether this most admirable theory, which 

 though abstruse is indispensable, was the work of Euclid himself, or a 

 predecessor, cannot now be known. The introduction of any definitely 

 numerical definition of proportion is rendered inaccurate by the 

 necessity of reasoning on quantities between which no exact numerical 

 ratio exists ; for which see INCOMMENSUHABLES. The method of 

 Euclid avoids the error altogether, by laying down a definition which 

 applies equally to commensurables and incomrnensurables, so that it is 

 not even necessary to mention this distinction. In the article PROPOR- 

 TION we shall endeavour to show that this method is more simple than 

 is generally supposed, and also that all substitutes for it have failed in 

 rigorous deduction. 



Book vi. applies the theory of proportion to geometry, and treats of 

 similar figures, that is, of figures which differ only in size, and not 

 in form. 



Book vii. lays down arithmetical definitions ; shows how to find the 

 greatest common measure and least common multiple of any two 

 numbers ; proves that numbers which are the least in any ratio arc 

 prime to one another, &c. 



Book viii. treats of continued and mean proportionals, showing when 

 it is possible to insert two integer mean proportionals between two 

 integers. 



Book ix. treats of square and cube numbers, as also of plane and 

 solid numbers (meaning numbers of two and three factors). It also 

 continues the consideration of continued proportionals, and of prime 

 numbers, shows that there is an infinite number of prime numbers, 

 and demonstrates the method of finding what are called perfect 

 numbers. 



Book x. contains 117 propositions, and is entirely filled with the 

 investigation and classification of certain incommensurable quantities. 

 It shows how far geometry can proceed in this branch of the subject 

 without algebra ; and though of all the other books it may be said that 

 they remain at this time as much adapted for instruction as when they 

 were written, yet of this particular buok it must be asserted that it 

 should never be read except by a student versed in algebra, and then 

 not as a part of mathematics, but of the history of mathematics. In 

 the article IRRATIONAL QUANTITIES we shall translate the phrases of 

 Euclid into algebraical language, by means of which we have no doubt 

 that many students will be enabled to read the book of Euclid with 

 profit. The book finishes with a demonstration that the side and 

 diagonal of a square are incommensurable. From this book it is most 



