UC-NRLF 
 
 $B m? 371 
 
 DE MORGAN. 
 
 CONNEXION OF NUMBER 
 
 MAGNITUDE. 
 
 Price 4s. 
 
 
PAULINE FORE MOFFITT 
 LIBRARY 
 
 UNIVERSITY OF CALIFORNIA 
 GENERAL LIBRARY, BERKELEY 
 
 i 
 
^r 
 
 Digitized by the Internet Archive 
 
 in 2007 with funding from 
 
 IVIicrosoft Corporation 
 
 http://www.archive.org/details/connexionofnumbeOOdemorich 
 

 THE 
 
 CONNEXION 
 
 OF 
 
 NUMBER AND MAGNITUDE 
 
 AN ATTEMPT TO EXl^LAIN 
 
 THE FIFTH BOOK OF EUCLID. 
 
 BY 
 
 AUGUSTUS DE MORGAN, 
 
 OF TRINITY COLLEGE, CAMBRIDGE. 
 
 La seule mani^re de bien trailer les 616mens d'une science exacte et rigoureuse, c'est d'y 
 mettre toute la rigueur et I'exactitude possible. — D'Alembsrt. 
 
 LONDON: 
 
 PRINTED FOR TAYLOR AND WALTON, 
 
 BOOKSELLERS AND PUBLISHERS TO THE UNIVERSITY OF LONDON, 
 
 30 UPPER GOWEE STREET. 
 
 M.DCCC. XXXVI. 
 
LONDON: 
 
 J. MOYES> CASTLE STKBET, JLBICESTEB SgUARE. 
 
 GIFT 
 

 PREFACE. 
 
 This Treatise is intended ultimately to form part of one 
 on Trigonometry. The place in which most students con- 
 sider number and magnitude together for the first time, is 
 in the elements of the latter science, unless they have un- 
 derstood the Fifth Book of Euclid better than is usually the 
 case. Previously, therefore, to commencing Trigonometry, 
 I consider it advisable to enter upon the consideration of 
 proportion in its strict form ; that is, upon the Fifth Book 
 of Euclid. There is no other method with which I am 
 acquainted which gives any thing like demonstration of the 
 general properties of ratios, though there is a dotix oreiller 
 pour reposer une tete Men faite, which many of the con- 
 tinental mathematicians have agreed shall be called demon- 
 stration, and which is beginning to make its way in this 
 country. 
 
 Hitherto, however, it has been customary for mathe- 
 matical students among us to read the Fifth Book of Euclid; 
 frequently without understanding it. The form in which it 
 appears in Simson's edition is certainly unnecessarily long, 
 and the tedious repetition of " A B is the same multiple of 
 CD which EF is of G H," in all the length of words, 
 renders the reasoning not easy to follow. The use of 
 general symbols of concrete magnitude, instead of the straight 
 line of Euclid, and of a general algebraical symbol for whole 
 
 307 
 
IV PREFACE. 
 
 number, seems to me to remove a great part of the difficulty. 
 Throughout this work it must be understood, that a capital 
 letter denotes a magnitude ; not a numerical representation, 
 but the magnitude itself: while a small letter denotes a 
 number, and mostly a whole number. And by the term 
 arithmetical proportion^ when it occurs, is signified, not the 
 common and now useless meaning of the words, but the 
 proportion of two magnitudes which are arithmetically re- 
 lated, or which are coinmensurable. 
 
 The subject is one of some real difficulty, arising from 
 the limited character of the symbols of arithmetic, con- 
 sidered as representatives of ratios, and the consequent 
 introduction of incommensurable ratios, that is, of ratios 
 which have no arithmetical representation. The whole 
 number of students is divided into two classes : those who 
 do not feel satisfied without rigorous definition and de- 
 duction ; and those who would rather miss both that take a 
 long road, while a shorter one can be cut at no greater ex- 
 pense, than that of declaring that there shall be propositions 
 which arithmetical demonstration declares there are not. 
 This work is intended for the former class. 
 
 AUGUSTUS DE MORGAN. 
 London f May 1, 1836. 
 
CONNEXION 
 
 or 
 
 NUMBER AND MAGNITUDE. 
 
 ERRATA. 
 
 Page 35, line 3, for less read greater. 
 
 47, ... \, for v{Q + Z) read w{Cl + Z). 
 
 — ... 2y for vQ read tvQ. 
 
 measureilieui 01 umugies^, wmun, m lut; wiucsi seuacj iii\ji.xx\xxj'a «.» 
 
 the applications of algebra to geometry, it will be right to inquire on 
 what sort of demonstration we are to pass from an arithmetical to a 
 geometrical proposition, or vice versa. 
 
 Geometry cannot proceed very far without arithmetic, and the 
 connexion was first made by Euclid in his Fifth Book, which is so 
 difficult a speculation, that it is either omitted, or not understood by 
 those who read it for the first time. And yet this same book, and the 
 logic of Aristotle, are the two most unobjectionable and unassailable 
 treatises which ever were written. 
 
 The reason of the difficulty which is found in the Fiflh Book is 
 twofold. Firstly; — It is all reasoning, unhelped by the senses: most 
 of the propositions have no portion of that intrinsic evidence which is 
 seen in " two sides of a triangle are greater than the third ; " but, at 
 the same time, the propositions of arithmetic which correspond to 
 
IV PREFACE. 
 
 number, seems to me to remove a great part of the difficulty. 
 Throughout this work it must be understood, that a capital 
 letter denotes a magnitude ; not a numerical representation, 
 but the magnitude itself: while a small letter denotes a 
 number, and mostly a whole number. And by the term 
 arithmetical proportion, when it occurs, is signified, not the 
 common and now useless meaning of the words, but the 
 proportion of two magnitudes which are arithmetically re- 
 lated, or which are commensurable. 
 
 The subject is one of some real difficulty, arising from 
 the limited character of the symbols of arithmetic, con- 
 
 which arithmetical demonstration declares there are not. 
 This work is intended for the former class. 
 
 AUGUSTUS DE MORGAN. 
 London f May 1, 1836. 
 
CONNEXION 
 
 OF 
 
 NUMBER AND MAGNITUDE. 
 
 When a student has acquired a moderate knowledge of the operations 
 and principles of algebra, with as many theorems of geometry as are 
 contained in the first four books of Euclid's Elements, it becomes 
 most desirable that he should gain some more exact knowledge of the 
 connexion between the ideas which are the foundation of one and 
 the other science, than would present itself either to an inattentive 
 reader, or to one whose whole attention is engrossed by the difficulty 
 of comprehending terms which cannot yet have become familiar to 
 him. Before proceeding, therefore, to explain Trigonometry (the 
 measurement of triangles), which, in the widest sense, includes all 
 the applications of algebra to geometry, it will be right to inquire on 
 what sort of demonstration we are to pass from an arithmetical to a 
 geometrical proposition, or vice versa. 
 
 Geometry cannot proceed very far without arithmetic, and the 
 connexion was first made by Euclid in his Fifth Book, which is so 
 difficult a speculation, that it is either omitted, or not understood by 
 those who read it for the first time. And yet this same book, and the 
 logic of Aristotle, are the two most unobjectionable and unassailable 
 treatises which ever were written. 
 
 The reason of the difficulty which is found in the Fifth Book is 
 twofold. Firstly; — It is all reasoning, unhelped by the senses: most 
 of the propositions have no portion of that intrinsic evidence which is 
 seen in " two sides of a triangle are greater than the third ; " but, at 
 the same time, the propositions of arithmetic which correspond to 
 
2 CONNEXION OF 
 
 tliose of the Fifth Book are very evident, and the student is therefore 
 led to escape from the notion of magnitude, and fly to that of number. 
 Secondly; — The non-existence of any very easy notation and system of 
 arithmetic in the time of Euclid, made geometrical considerations re- 
 latively so much more simple, that the form of his book is (to us) 
 unnecessarily remote from all likeness to a treatise connected with 
 numbers. The difference between our day and his lies in tiiis : that 
 in the former the exactness of geometry was gained with some degree 
 of prolixity and (to a beginner) obscurity ; in the latter, the facility 
 of arithmetic is preferred, and perfect demonstration is more or less 
 sacrificed to it. 1 shall now endeavour to present the Fifth Book of 
 Euclid in a form which will be more easy than tlie original, to those 
 who have some acquaintance with algebra. 
 
 By number is here meant what is called abstract number, which 
 merely conveys the notion of limes or repetitions, considered inde- 
 pendently of the things counted or repeated. By magnitudey or 
 quantity, is meant a thing presented to us, not as to its form, if it 
 have form, or as to colour, weight, or any other circumstance, but 
 simply as that which is made up of parts, not differing from the whole 
 in any thing but in being less ; so that, if we consider separately a 
 part and the whole, we have only two inferences : 
 The part is less than the whole. 
 The whole is greater than the part. 
 
 Every thing we can see or feel presents to us the notion of mag- 
 nitude or quantity. And here we must observe, that we have to pick 
 our words from among those in common use, which never have very 
 precise meanings. For instance, we have magnitude, the nearest 
 English word to which is greatness ; and quantity, for which the 
 word, if it existed, should be so-much-ness. These words are of the 
 same meaning, and the more indefinite we now leave them (except 
 only in assigning that they are to be considered as applied to any 
 thing which can be made more or less), the better for our purpose ; 
 since it is the object of this treatise to deduce from that indefinite 
 notion a method of making mathematical comparisons of quantities, 
 by aid of the notion of number. 
 
 Upon two magnitudes, our senses will enable us to draw one or 
 other of the following conclusions : 
 
 1. The first is sensibly greater than the second. 
 
 2. The first is sensibly less than the second. 
 
NUMBER AND MAGNITUDE. 6 
 
 3. The first is sensibly equal to the second ; meaning that the dif- 
 ference, if any, is so small that our senses cannot perceive it. This is 
 what is meant by equality of magnitudes in common life. The English 
 foot and the Florence foot are equal for common purposes : they 
 differ by about the twentieth part of an inch, which in a foot is called 
 nothing. 
 
 Perfect equality is a mathematical conception, which never can be 
 absolutely verified in practice ; for so long as the senses cannot per- 
 ceive a certain quantity, be it ever so small, so long it must always be 
 possible that two quantities, which appear equal, may differ by as 
 much as the imperceptible quantity. But we are not reasoning upon 
 what we can carry into effect, but upon the conceptions of our own 
 minds, which are the exact limits we are led to imagine by the rough 
 processes of our hands. The following, then, is the postulate upon 
 which we construct our results : 
 
 Ani/ one magnitude being given, let it be granted that any number 
 of others may be found, each of which is {positively and mathematically) 
 equal to the first. 
 
 Let A represent a magnitude — not as in algebra, the number of 
 units which it contains, but the magnitude itself — so that if it be, for 
 instance, weight of which we are speaking, A is not a number of 
 pounds, but the weight itself. Let B represent another magnitude of 
 the same kind ; we can then make a third magnitude, either by putting 
 the two magnitudes together, or by taking away from the greater a 
 magnitude equal to the less. Let these be represented by A -j- B and 
 A — B, A being supposed the greater. We can also construct other 
 magnitudes, by taking a number of magnitudes each equal to A, and 
 putting any number of them together. Thus we have 
 
 A + A which abbreviate into 2 A 
 
 A-l-A + A 3A 
 
 A + A-j-A + A 4A 
 
 and so on. We have thus a set of magnitudes, depending upon A, 
 and all known when A is known ; namely, 
 
 A 2A 3A 4A 5A &c. 
 
 which we can carry as far as we please. These (except the first) are 
 distinguished from all other magnitudes by the name oi multiples of A ; 
 and it is evident that they increase continually. Let the preceding be 
 
4 CONNEXION OF 
 
 called the scale of multiples of A. It is clear that the multiples of mul- 
 tiples are multiples ; thus, 7 times 3 A is 21 A, in times nA is (w?n)A, 
 where mn is the arithmetical product of the whole numbers m and n. 
 The following propositions may then be proved. 
 
 Prop. I. If A be made up of B and C, then any multiple of A 
 is made up of the same multiples of B and C ; for 2 A must be made 
 up of 
 
 B C B C 
 
 of which B and B make 2 B, C and C make 2 C ; so that 2 A is made 
 lip of 2B and 2C. Similarly, 3 A is made up of 
 
 B C B C B C 
 
 or of 3B and 3C. 
 
 Corollary. Hence it follows, that if A be less than B by C, any 
 multiple of A is less than the same multiple of B by the same mul- 
 tiple of C. For, since A is less than B by C, A and C together make 
 up B ; therefore, 2 A and 2 C make up 2 B, or 2 A is less than 2B by 
 2 C. The algebraical representations of these theorems are as follows : 
 
 If A = B-|-C mA = mB + ?wC 
 
 If A = B-C 7wA = mB-?wC 
 
 m being any of the numbers 2, 3, 4, &c 
 
 Prop. II. However small A may be, or however great B may 
 be, the multiples in the scale 
 
 A, 2 A, 3 A, 4 A, 5 A, &c. 
 
 will come in time to exceed B, by continuing the scale sufficiently 
 far : B and A being magnitudes of the same kind. This is a pro- 
 position which must be considered as self-evident : it must be re- 
 membered that B remains the same, while we pass from one multiple 
 of A to the next. Put feet together and we shall come in time to 
 exceed any number of miles, say a thousand. But the best illustration 
 of the reason why we formally put forward so self-evident a pro- 
 position, will be to remark, that it is not every way of adding mag- 
 nitude to magnitude without end, which will enable us to surpass 
 any given magnitude. To a magnitude add its half; to that sum add 
 half of the half; to which add the half of the last: and so on. No 
 continuation of this process, were it performed a hundred million of 
 times, could ever double the first magnitude. 
 
NUMBER AND MAGNITUDE. O 
 
 Prop. III. If A be greater than B, any multiple of A is greater 
 than the same multiple of B. This follows from Prop. I. And if A 
 be less than B, any multiple of A is less than the same multiple of B. 
 This follows from the corollary, Prop. I. And if A be equal to B, 
 any multiple of A is equal to the same multiple of B. This is self- 
 evident. 
 
 Prop. IV. If any multiple of A be greater than (equal to, or less 
 than) the same multiple of B, then A is greater than (equal to, or 
 less than) B. For example, let 4 A be greater than 4 B ; then A 
 must be greater than B; for, if not, 4A would be equal to, or less 
 than, 4 B (Prop. lit.). 
 
 Prop. V. If from a magnitude the greater part be taken away; 
 and if from the remainder the greater part of itself be taken away, and 
 so on : the given magnitude may thus be made as small as we please, 
 meaning as small as, or smaller than, any second magnitude we 
 choose to name. 
 
 Let A and Z be the two magnitudes, and let A diminished by 
 more than its half be B, then 2 B is less than A. Let B diminished 
 by more than half be C ; then 2C is less than B, 4C is less than 2 B, 
 and still more less than A. Let C diminished by more than its half 
 be D, then 2 D is less than C, 8 D is less than 4 C, and still more 
 than A. This process must end by bringing one of the quantities 
 
 A, B, C, D, &c. below Z in magnitude. For, if not, let A, B, C, &c. 
 always remain greater than Z. Then, since 2B, 4C, 8 D, 16E, &c. 
 are all less than A (just proved) still more must 2Z, 4Z, 8Z, 16Z, 
 &c. be less than A. But this cannot be ; therefore, one of the set A, 
 
 B, C, &c. must be less than Z. 
 
 [The reductio ad absurdum, as this sort of argument is usually 
 called, is a difficult form of a simple inference. Suppose it proved 
 that whenever P is Q, then X is Y. It follows that whenever X is 
 not y, P is not Q. It is usually held enough to say, for if P were Q 
 X would be Y. But the form in which Euclid argues, supposes an 
 opponent ; and the whole argument then stands as follows. " When 
 X is Y, you grant that P is Q ; but you grant that P is not Q. I say 
 that X is not Y. If you deny this you must affirm that X is Y, of 
 which you admit it to be a consequence that P is Q. But you grant 
 that P is not Q ; therefore, you say at one time that P is Q and that 
 P is not Q. Consequently, one or other of your assertions is wrong, 
 
 B 2 
 
S CONNEXION OF 
 
 either ' P is not Q' or ' X is Y/ If the first be right, the second 
 is wrong : lliat is, ' X is not Y ' is right." 
 
 The preceding argument runs as follows ; — when A, B, C, &c. are 
 all greater than Z, then 2Z, 4Z, &c. are all less than A: but 2Z, 
 4 Z, &c. are not all less than A ; therefore, A, B, C, &c. are not all 
 greater than Z]. 
 
 Corollary, The preceding proposition is equally true when, 
 iistead of taking more than the half at each step, we take the half 
 itself in some or all of the steps. 
 
 Prop. VI. If there be two magnitudes of the same kind, A and 
 B, and if the scales of multiples be formed 
 
 A, 2 A, 3 A, &c. B, 2B, 3B, &c. 
 
 then one of these two things must be true; either, there are mul- 
 tiples in the first scale which are equal to multiples in the second 
 scale ; or, there are multiples in the first scale which are as nearly- 
 equal as we please to multiples (not the same perhaps) in the second 
 set: that is, we can find one of the first set, say m A, which shall 
 either be equal to another in the second set, say nB, or shall exceed 
 or fall short of it by a quantity less than a given quantity Z, which we 
 may name as small as we please. 
 
 Let us take a multiple out of each set, any we please, say pA. and 
 gB. IfjsAandjBbe equal, the first part of the alternative exists; 
 if not, one must exceed the other. Let p A exceed q B, say by E ; 
 then we have 
 
 ;?A = ^B + E (1) 
 
 Now E is either less than B, or equal to B, or greater than B. If 
 tlie first, let it remain for the present; if the second, we have 
 pA = (9- + 1) B, or the first alternative exists : if the third, then B 
 can be so multiplied as to exceed E. Let (^ + 1) B be the first 
 multiple of B which exceeds E ; that is, let the next below, or t B, 
 be less than E, say by G, then we have 
 
 E=^B + G pA = ^B + ^B + G 
 
 or j?A = (^ + OB + G 
 
 Now G must be less than B ; for E or ^B + G is less than (^ + 1) B, 
 or ^B + B. We have then made this first step (observe that q ■}- t'ls 
 
NUMBER AND MAGNITUDE. 7 
 
 only so7ne multiple of B; call itrB). Either the first alternative 
 exists, or we can find p A and r B, so that 
 
 p A = r B + G where G is less than B (2) 
 
 Now G can be so multiplied as to exceed B ; let uG and (v + 1) G 
 be the multiples of G, between which B lies, so that 
 
 ?;G is less than B, say 2J G = B — K 
 (v + \)Q is greater than B, say ?; G + G = B + L 
 
 and it follows that K4-L = G;for since (v+l)G and vG differ 
 by G, if a magnitude lie between them, their difference must be made 
 up of the excess of that magnitude over the lesser, together with its 
 defect from the greater. Consequently, either K and L are both 
 halves of G, or one of them falls short of the half. Suppose K is less 
 than the half of G : then take both sides of (2), v times, and we have 
 
 t;^A = v/'B+vG 
 or zjpA = z?rB + B — K 
 
 or vpA. = (y r -f I) B — K (K less than half G) 
 
 But if L be less than the half of G, take both sides of (2) t; + 1 times 
 which gives 
 
 (y + l)pA = (v-hl)rB+(v+\)G 
 or {v+l)pA= zTTi-rB + B + L 
 or V+l pA = (v+1 7* + 1)B + L (L less than half G) 
 
 If K and L are both halves of G, we may take either. And if (a case 
 not yet included) a multiple of G, uG, be exactly equal to B, we 
 have then 
 
 vpA = vrB + vG = (vr + 1) B 
 
 which gives the first alternative. Consequently, we either prove the 
 first alternative, or we reduce the equation 
 
 pA = rB + G (G less than B) 
 
 to an equation of the form 
 
 p'A = r'B ± G' ^G' not greater than the half of G. 
 
 We may now proceed as before ; but, to exemplify all the cases that 
 may arise, let us take 
 
(v' + l)G'= B + L'i 
 
 p"A = /A±G"| 
 
 8 CONNEXION OF 
 
 p'A =:/B~G' 
 
 If y'G' be exactly B, we prove the first alternative, as before ; but if 
 B lie between v'G' and (v'+l) G', let us suppose 
 
 and K'+L'= G' 
 as before, 
 
 in which one of the two, K' or L', will not be greater than the half of 
 G', so that we obtain by the same process, an equation of the form 
 
 G" not greater than 
 the half of G'. 
 
 By proceeding in this way, we prove either, 1. The first alternative 
 of the proposition; or, 2. the possibility of forming a continued set 
 of equations 
 
 pA=qB±G, pA = q'B±G', p"A = q"B±G'\ &c. 
 
 where, in the scale of quantities G, G', G", &c., no one exceeds the 
 half of the preceding. Consequently, we may (unless interrupted by 
 the first alternative) carry on this process until one of the quantities 
 G, G', G" &c. is smaller than Z (Prop. V.) that is, we have either 
 the first or second alternative of the problem. And exactly the same 
 demonstration may be applied to the case, where at the outset 
 j!>A = ^B — E. 
 
 This proposition proves nothing of a single magnitude, but it 
 establishes two apparently very distinct relations between magnitudes 
 considered in pairs. Tliere may be cases in which the first alternative 
 is established at last : and there may be cases in which it is never 
 established. We shall first take the case in which the first alternative 
 is established. 
 
 Suppose it ascertained by the preceding process that 
 
 8A = 5B 
 
 Here is an arithmetical equation between the magnitudes : and there- 
 fore any processes of concrete arithmetic will apply. Take the 
 40th part (8x5 = 40) of both sides, 
 
 ,. , . 8A 5B A B 
 
 which gives 40 ="4^ °^ 5=8- 
 
 consequently the fifth part of A is the same as the eighth part of B, 
 or that which is contained 5 times in A is also that which is contained 
 
NUMBER AND MAGNITUDE. ^ 
 
 8 times in B. Let this fifth of A or eighth of B be called M; then 
 A = 5M, B = 8M, and A and Bare both multiples ofM. Con- 
 sequently, when the first alternative of Prop. VI. exists, both A and 
 B are multiples of some third magnitude M. The converse is readily 
 proved, namely, that when A and B are both multiples of any third 
 magnitude, the first alternative of Prop. VI. is true. For if A=j:M, 
 B=3/M, we have yA = yjrM, xB=zxi/Mt or i/A = xB. The 
 term measure is used conversely to multiple, thus : if A be a multiple 
 of M, M is said to be a measure of B. Hence in the case we are 
 now considering, A and B have a common measure^ and are said to 
 be commensurable. We have therefore shewn that all commensur- 
 able magnitudes, and commensurable magnitudes only, satisfy this 
 first alternative. 
 
 There remains, then, only the second case to consider, which it is 
 now evident contains those magnitudes (if any such there be) which 
 have no common measure whatsoever. The question therefore is, 
 Are there such things as incommensurable magnitudes? On this 
 point the second alternative shews that our senses cannot judge, for 
 let Z be the least magnitude of the kind in question, which they are 
 capable of perceiving (of course with the best telescopes, or other 
 means of magnifying small quantities which can be obtained) then 
 we know that;)A may be made to differ from ^B by less than Z, 
 that is, we may say that all magnitudes are sensibli/ commensurable. 
 But it evidently does not follow that all magnitudes are mathema- 
 tically commensurable; and it has been shewn, by process of de- 
 monstration, that there are * incommensurable quantities in such 
 abundance, that take almost any process of geometry we please, the 
 odds are immense against any two results being commensurable. 
 
 The suspicion that all magnitudes must be commensurable led to 
 the attempt, which lasted for centuries, to find the exact ratio of the 
 circumference of a circle to its diameter. And even now, though the 
 adventure is never tried by those who have knowledge enough to 
 read demonstration of its impossibility, no small number of persons 
 
 • Legendre, and others before him, have shewn that the diameter 
 and circumference of a circle are incommensurable ; and the student will 
 find in my Algebra, p. 98, or in the Lib. Useful Know., treatise on the 
 Study of Mathematics, p. 81, proof that the side and diagonal of a square 
 are incommensurables. Also in Legendre 's Geometry, or Sir D. Brew- 
 ster's Translation. 
 
10 CONNEXION OF 
 
 exercise themselves by endeavouring to make an elementary ac- 
 quaintance with geometry (and sometimes none at all) overcome this 
 difficulty. It is our business here to shew how strict deductions 
 may be made upon quantities which are incommensurable, with the 
 same facility, and in the same manner, as upon commensurables. If 
 we call any length (say that known by the name of a foot), the unit 
 of its kind, and denote it in calculation by 1, we must call twice such 
 a magnitude 2, and so on ; half such a magnitude i, and so on. We 
 may then apply arithmetic, every possible subject of which is con- 
 tained in the following infinitely extended table. 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 2 
 
 &c. 
 
 
 
 
 1 
 2 
 
 
 
 1 
 
 
 3 
 2 
 
 
 
 2 
 
 &c. 
 
 
 
 1 
 
 3 
 
 
 2 
 3 
 
 
 1 
 
 4 
 3 
 
 
 5 
 3 
 
 
 2 
 
 &c. 
 
 
 
 1 
 
 4 
 &c. 
 
 2 
 
 4 
 
 
 3 
 4 
 &c. 
 
 1 
 
 5 
 4 
 &C. 
 
 6 
 4 
 
 
 7 
 4 
 &c. 
 
 2 
 
 &c. 
 
 And every length which is commensurable with the foot is in- 
 cluded, in many different forms, in this table. Let F represent the 
 foot, L any other length, let M be their common measure : let 
 
 F=/M, L = ZM, then ZF=/M, or L 
 
 L = 7; F = J. when F is called 1. 
 
 But it is plain that we cannot, by any arithmetic of length founded 
 upon the foot as a unit, draw conclusions as to lengths which are in- 
 commensurable with the foot, though we can perhaps do so for any 
 practical purpose. Let L be a length which is such, and let Z be a 
 length so small as to be immaterial for the purpose in question. Then, 
 we can determine I andy, so that 
 
 /L=ZF±G (G less than Z) 
 
 L = iF±J: 
 
 so that, by assuming L = -7.F, we commit an error, in excess or defect, 
 
 less than G, and therefore immaterial. With such a process many 
 minds would rest contented : but there is a consideration which will 
 
NUMBER AND MAGNITUDE. 1 1 
 
 Stand in the way of perfect satisfaction, or at least ought to do so. 
 Granting that in the preceding case the error at the outset is imma- 
 terial, let us suppose the student disposed to substitute for all incom- 
 mensurables, magnitudes very near to them which are commensurables, 
 and thus to continue his career till he comes to the highest branches 
 of applied mathematics. Let us suppose a set of processes, beginning 
 in arithmetic, continued through algebra, the differential calculus, &c., 
 up to a point in optics or astronomy, in a series of results, embracing, 
 we may suppose, ten thousand inferences. If he set out with an 
 erroneous method, what security has he that the error will not be mul- 
 tiplied ten thousand fold at the end, and thus become of perceptible 
 magnitude. If somebody acquainted with the subject have told him 
 that it will not so happen, he might as well skip the intermediate 
 sciences and receive the result he wants to obtain on the authority of 
 that person, as study them in a manner, the correctness or incorrectness 
 of which depends on that person's authority. If he answer that the 
 result, namely, such multiplication of errors, appears extremely im- 
 probable, it may be replied, firstly., that that is more than he can 
 undertake to decide ; secondly, that by pursuing his mathematical 
 studies on such a presumption, he makes all the pure sciences present 
 probable results only, not demonstrated results ; more probable, per- 
 haps, than many parts of history, but resting on an impression which 
 must, in his mind, be the result of testimony. 
 
 It appears, however, that we may expect series of collateral results, 
 the one for commensurables, the other for incommensurables, and 
 presenting great resemblances to each other ; for we may, by any 
 alteration, however minute, convert the latter kind of magnitude into 
 the former. But this we may prevent, by extending our notions of 
 arithmetical operations, or rather by applying to magnitude processes 
 which are usually applied to number only, as follows : 
 
 If we examine the processes of arithmetic, we find, 1st, Addition 
 and substraction, to which abstract number is not necessary, since the 
 concrete magnitudes themselves can be added or subtracted. 2d, Mul- 
 tiplication, the raising of powers and the extraction of roots, in all of 
 which abstract number is essentially supposed to be the subject of 
 operation. 3d, Division, in which it is not necessary to suppose 
 abstract number in finding the whole part of the quotient, but in 
 which we cannot, without reference to numbers, compare the remain- 
 der and divisor, in order to form the finishing fraction of the quotient. 
 
12 CONNEXION OF 
 
 4th, The process of finding the greatest common measure of two quan- 
 tities, in which the remainder is not compared with the divisor, except 
 in a manner which is as applicable to the case of concrete magnitudes 
 as of abstract numbers. To shew this, we shall demonstrate the me- 
 thod of finding the greatest common measure of two magnitudes. 
 
 Let A and B be two magnitudes, which have a common measure 
 M ; let A = a M, B = i M . Then, it is clear that 
 
 a;A + ?/B or ixa+yh)M, a:A — ?/B or (j;a--yZ>)M 
 have the same measure, unless it should happen that in the latter case 
 aro = i/6, in which case xA=3/B. Let A be the greater of the two, 
 and let A contain B more than /5 and less than /3 -f- 1 times, so that 
 A=/3B + B', when B' is less than B. Then B' being A — /3B, is 
 measured by M. Let B contain B' more than p>' and less than )3' + 1 
 times ; or let B=/5'B'+ B'' where B" is less than B'. Let B' contain 
 B" more than /S" times, &c., or let B'=/3"B" -f B'", and so on. And 
 B" or B — /i'B'is measured by M, &c. We have then the following 
 
 conditions : 
 
 A is a multiple of M 
 
 B M 
 
 A = i8 B + B' B' < B, but is a multiple of M 
 
 B =/3'B + B" B" <B' 
 
 B'=/3"B'+B"' B' <B'' 
 
 Now, since B B' B" are decreasing quantities, and all multiples 
 of M, they are all to be found in the series, 
 
 M, 2M, 3M, 4M, &c. 
 
 in which continual decrease must bring us at last to nothing, or we 
 must end with an equation of the form.* 
 
 that is, one remainder is a multiple of the next. To take a case, let 
 the fifth equation finish the process ; so that, in addition to the pre- 
 ceding, we have 
 
 B" = /3"'B"' H- B^^ 
 B'" = /3'^B'^ 
 
 * When a letter denotes an indefinite number of accents, it is dis- 
 tinguished from an exponent by being placed iu brackets, and higher 
 numbers of accents than three are usually denoted by Roman numerals. 
 
NUMBER AND MAGNITUDE. 13 
 
 In the fourth, substitute B'" from the fifth, giving 
 
 In the third, substitute B" and B'" as found, giving 
 B' = (/3" /3"; /3'^ + (3" + (3^') B^^ 
 
 In the second, substitute B' and B" as found, giving 
 
 B = ((3' ^'^'"(3^'' + /3'/3" + /3'/3^^ + ^'"/3^^ + 1) B^^ 
 In the first, substitute B and B' as found, giving 
 
 Consequently, Bi^ is a common measure of A and B; but, since M at 
 the outset is any common measure we please, let it be the greatest 
 common measure. Then B^^ must be M, for it is in the series M, 
 2 M, &c. ; and were it any other than M, there would be B*'' a com- 
 mon measure, greater than the greatest. Hence this process deter- 
 mines the greatest common measure, and also the number of times 
 which each of the two, A and B, contains the greatest common 
 measure. 
 
 It is here most essential to observe, that this whole process is in- 
 dependent of any arithmetic, except pure addition and subtraction, 
 which can be performed on the magnitudes themselves, without any 
 numerical relation whatsoever; the only thing required being the 
 axiom in page 3. We shall actually exemplify this on two right 
 
 lines. 
 
 X y 
 
 A h -^ ♦ '-\ 
 
 B h ♦ 1 
 
 z k 
 
 A = B + (xij) B = {xi/) + (zk) {xy) = 2{zk) 
 Therefore B = ^{zk) A = 6{zk) 
 
 In this case, by actual measurement (supposed geometrically exact) 
 B and A are found to be respectively 3 and 5 times zk. 
 
 When the preceding process has an end, we therefore detect the 
 greatest common measure ; and we have shewn, that where there is a 
 common measure, the process must give it, with the converse. Con- 
 sequently, in the case where there is no common measure, this process 
 must go on for ever, and we have an interminable series of equations, 
 A=:/3BH-B', B=/3'B'-i-B", B'=^"B"+B'", &c., the conditions of 
 
 c 
 
14 CONNEXION OF 
 
 which are, tliat B, B', B", B'", &c., are a continually decreasing 
 series, though it does not follow that each one was less than the half 
 of the preceding. We shall now examine the effect of successive sub- 
 stitutions from the beginning, first making the following remark : If 
 there be any two incommensurable quantities A and B, of which A is 
 the greater, then there follows an interminable set of whole numbers, 
 /S, iS', /3", .... which are not subject to any particular law, but can 
 be found when A and B are given ; and an interminable set of quan- 
 tities, A, B, B', B", .... connected with the former by this law, that 
 A contains B between /3 and /3 -[- 1 times ; B contains B' between /S' 
 and /S'+l times, and so on. 
 
 We have B' = A — jSB 
 
 B" = B-/3'B' = B-^ (A-^B) 
 or . B" = (/3/3'+l)B-/3'A 
 
 B"'= B'-/3"B" = A-^B-(/3/3' + l)/3"B + /5'/r'A 
 = (/3'/3"-hl)A-(/3/3'/3" + /3-F/3")B 
 
 and thus we go on representing the remainders alternately, in the form 
 pA — qB and qB — pA. We may easily find the law of the co- 
 efficients, as follows : 
 Suppose we come to 
 
 B("> = qB-pA 
 B(n+i) = p'A-r/B 
 Then we have B^") = /3("+i) B<"+i) -f- B<"+2) 
 or B('»+2> = B<"> — /3("+^) B(»+i) 
 
 = ^B-pA-|8("+i) (p'A-q'B) 
 = {l3("+^)q' + q)B-{l3<''+'^p'+p)A 
 or if, continuing the preceding notation, we suppose 
 
 we have f = /3("+i) p -h p q" = /3("+'> q' + q 
 
 so that, if we write the values of B', B", .... with the following no- 
 tation, putting opposite to each the /3(**) which occurs for the first time 
 in the B(") of the equation ; namely, 
 
 B' =;?iA-^iB ^ 
 
 B" = q^B-^p^A ^ 
 
NUMBER AND MAGNITUDE. 15 
 
 B'" ^p,A-^q,B ^' 
 
 B'^ =q^B^p^A ^"' 
 
 &c. &c. &c. 
 
 we have the following uniform method of forming pn and qn for dif- 
 ferent values of n in succession. 
 
 i?i = 1 ^1 = /3 
 
 po = /3' ^2 = /3'/3 + 1 
 
 Ps = /3>2 -\-pi 93 = {^'92 + 5'l 
 
 &c. &c. &c. See. 
 
 in which it is plain, from the method of formation, that p^ p^ &c. 
 y, ^2 &c. are increasing whole numbers, so that we may continue, 
 supposing B' B" .... never fail, till pn and qn are greater than any 
 number named. And since B' B'' .... are all less than B, and 
 therefore less than A, we have the following succession of results, 
 ad infinitum. 
 
 pi A is greater than ^'iB but less than (§'i -f 1)B 
 
 p^A is less than q^B but greater than (q^ — 1)B 
 
 p^A is greater than ^'sB but less than {q^+l)^ 
 
 &c. &c. &c. 
 
 Hence, it appears that 
 
 A is greater than 
 
 less than 
 
 greater than 
 
 less than — B &c. ad inf. 
 
 Now, from this table of relations, we can determine whether any 
 given multiple of A, ^A, is greater or less than any given multiple of 
 B, 1/ B. To do this we must inquire between what two consecutive 
 multiples of B does xA lie. 
 
 We now proceed as follows : 
 
 1. We must shew that any fraction, such as 
 
 -7— — lies between r and — 
 b+n n 
 
 1l 
 
 Pi 
 
 B 
 
 1± 
 
 B 
 
 Pz 
 
 B 
 
16 CONNEXION OF 
 
 unless where the two latter are equal, in which case the first also is 
 the same. The preceding must be true if 
 
 a + m lies between ^ ( J -f w) and - (b + n) 
 
 or a -h -r- and — -4- m 
 
 b n 
 
 , , on hm 
 
 or a + 7w + __wiand a+m+ a 
 
 n 
 
 am -, m I a 
 
 or a + m+n-r — n- and a + m + b b-r 
 
 n n b 
 
 which is evidently true : for if - be greater than -, the first is greater 
 
 than a + in, and the second less ; if ■=- be less than — , vice versa. 
 
 n 
 
 2. We now see that 
 
 - -^ £^ li^« between ^ and ^ - 
 Pa I^Pi+Pi A"i?2 Pi 
 
 ^^ and -^ 
 Pa Pi 
 
 1± or ^'"^^-^^^ Si Si 
 
 Pa /3">3+/>2 Pz Pi 
 
 and so on. Consequently, to arrange all the fractions thus con- 
 sidered, in order of magnitude, we must write them thus, 
 
 ll Si 2± Si Si Si 
 
 Pi Pi Ps Pe P4 P2 
 
 3 . We can thus bring two fractions as near together as we please : 
 to prove this, take three consecutive fractions 
 
 gm gm+i /gm+2 ^^ (^^"^ gm+1 +gm 
 
 or 
 
 ~tgm \ 
 
 -{■Pmj 
 
 Pm Pm+1 \pm+2 /3^'"+^>J9m+l + fn 
 
 which reduced to a common denominator, the first and second, and 
 the second and third, give 
 
 gm Pm+l gm+l Pm 
 
 Pm pm+1 Pm+1 fm 
 
 A (^^""^^^gm+l Pm+1 + Pm gm+l ^^^ ^^"^"^^^gm+l Pm+l + Pm+1 gm 
 Pm+1 Pm+2 Pm+l pm+2 
 
 in which it is clear that the difference of the numerators is the same 
 in each couple, but that if the first numerator be the greater of the 
 
NUMBER AND MAGNITUDE. 17 
 
 first couple, the second numerator is that of the second ; a result we 
 might have foreseen, having proved that 
 
 £^i±i lies between ^-^^ and ^. 
 
 Hence it follows that the numerator of the difference of any two 
 successive fractions of the set 
 
 9i_ go: 93 
 Pi P2 P3 
 
 is the same as that of the difference immediately preceding, that is, 
 the difference of — and ^^^^ has the same numerator as 
 
 Pn pn-l 
 
 the difference of ^^^ and ^"~^ which has the same 
 
 Pn-l Pn-2 
 
 numerator as the difference of — and — ; but 
 
 P2 Pi 
 
 9i ^2 _ 1 /3' _ 1 
 
 therefore this numerator of the differences is always 1, or 
 ?!L and ^-^^ differ by 
 
 Pn Pn+1 Pn Pn+1 
 
 Hence the difference may be made as small as we please, or smaller 
 than any fraction — named by us, since pn itself can be made greater 
 than m, much more pn pn+i- 
 
 4. These fractions cannot for ever lie alternately on one side and 
 
 V 
 
 the other of any given fraction -. 
 
 For if this were possible, then, since A lies between 
 ^JLB and ^^^B 
 
 Pn Pn+1 
 
 V 
 
 and since by the supposition -B does the same, and since the couple 
 
 just mentioned can be made to differ, by as small a fraction of B as 
 we please, then we should have 
 
 A = -B ± K 
 
 c 2 
 
18 CONNEXION OF 
 
 where K may be made as small as we please. Now this is saying 
 
 V 
 
 that A = - B ; for A must either be 
 
 X 
 
 -B or -B ih some definite magnitude; 
 but the latter it is not ; for the supposition we are trying leads to 
 
 A = - B + a magnitude as small as we please. 
 Consequently, our supposition that the series of fractions lie alter- 
 
 V 
 
 nately on one side and the .other of a definite fraction -, leads to the 
 
 X 
 
 conclusion that A and B are commensurable, or the process of finding 
 B' B" .... finishes, as we have shewn. But it does not finish, by 
 hypothesis; therefore the series of fractions cannot lie alternately 
 
 V 
 
 on one side and the other of -. 
 
 X 
 
 We can now shew between what multiples of B j^A must lie. It 
 is clear that 
 
 a: A lies between -^^B and ^Jlt±l^ : 
 
 Pn Pn+] 
 
 now it is not possible that any whole number v should always lie 
 
 between — ~ and ^""^^ ; for if so, then would 
 pn pn+i 
 
 - always lie between — and "" ■■- 
 
 •I" Pn pn+l 
 
 which has been proved to be impossible. Consequently, 
 
 ■^^"■R A '^9"+i"R (which approach each other 
 Pn Pn+i without limit) 
 
 must come at last always to lie between two multiples of B; and 
 still more must x A, which lies between the7n. Hence, by proceeding 
 far enough, we can always find between what multiples of B lies x A; 
 and thence whether xA is greater or less than 3^ B. 
 
 We have thus divided all pairs of magnitudes into two classes, 
 1. Commensurahles, in which we can always say that A = -B, 
 
 (J and p being whole numbers, and can always tell exactly by what 
 fraction of A or B, a- A exceeds or falls short of j/B. For we have 
 
NUMBER AND MAGNITUDE. 19 
 
 xA^yB= (a;-2/^)A= (a;^-?/)B if xA>yB 
 
 (yB-a;A)= (|y-^)A= (y-^|)B if a;A<yB 
 
 2. Incommensurables, in which we can never say A = - B, but 
 
 P 
 in which we can assign a series of fractions alternately increasing and 
 
 decreasing, but making less and less change at every step, 
 
 ll is ^ 
 
 Pi P2 Pa 
 
 and such that A is greater than — B, less than— B, &c. ad infinitum : 
 
 Pi P2 
 
 so that we can always assign 
 
 A = ^B + K 
 
 Pn 
 
 where K is less than any magnitude we name ; and such that we can 
 always tell by them whether ^A exceeds or falls short of z/B, but not 
 exactly how much. 
 
 Let us suppose, as an example, that we have two magnitudes A 
 and B, which tried by the process in page 13, give 
 
 A = B + B', B = B' + B ', B' = B" + B", &c. ad inf. 
 
 or suppose /3 = 1, /3' = 1, /3" = 1, &c. ad inf. 
 
 Hence the several values ofp and q are Pi = l, ^3 = 1, ^3 = 2, &c. 
 
 as in this table, 
 
 12 3 4 5 6 7 8 9 10 11 12 ^ 
 
 — — occ. 
 
 p 1 1 2 3 5 8 13 21 34 55 89 144 
 
 5- 1 2 3 5 8 13 21 34 55 89 144 233 &c. 
 
 a>b<2B>^b<|b>|b<Hb &c. 
 
 Hence A lies between B and 2B 
 
 2A 3B .. 4B 
 
 3A 4B .. 5B 
 
 4A 6B .. 7B 
 
 5A 8B .. 9B&C. 
 
 If we wish to know between what multiples of B 100 A lies, 
 
 we find 
 
20 CONNEXION OF 
 
 A> W^ < liB; lOOA > 16lgB < 161 li|B 
 
 or 100 A lies between 161 B and 162 B. 
 
 We can thus form what we may call a relative multiple scale made 
 by writing down the multiples of A, and inserting the multiples of 
 B in their proper places; or vice versa. In the instance just given 
 the commencement of this scale is 
 
 B, A, 2B, 3B, 2A, 4B, 3A, 5B, 6B, 4A, 7B, 8B, 5A, 9B, &c. 
 
 which we may continue as far as we please by simple arithmetic. If 
 the magnitudes in question be lines, we may represent this multiple 
 scale as follows : 
 
 01 1 — )fH B< 1 )H \-^ 1 ^ 1 
 
 Measuring from O, the crosses mark off multiples of A, and the bars 
 multiples of B. Thus 
 
 01i = B Ol2=2B Ol3=3B&c. 
 Oxi = A 0x2= 2A Ox3 = 3A&c. 
 
 We shall now proceed to some considerations connected with 
 a multiple scale, for the purpose of accustoming the mind of the 
 student to its consideration. We may imagine a scale like the pre- 
 ceding to be equivalent to an infinite number of assertions or nega- 
 tions, each one connected with the interval of magnitude lying between 
 two multiples of B. Thus, the preceding scale contains the following 
 list ad infinitum. 
 
 ^ 1. Between and B lies no multiple of A 
 
 2. Between B and 2B lies A 
 
 3. Between 2B and 3B lies no multiple of A 
 
 4. Between 3B and 4B lies 2 A 
 &c. &c. &c. &c. 
 
 Now, on this we remark, 1st, That the negatives of the above 
 series, though they appear at first to prove nothing, yet in reality 
 have each an infinite number of negative consequences. From the 
 third assertion of the preceding list, namely, neither A, nor 2 A, nor 
 3 A, &c. lies between 2B and 3B, we immediately deduce all the 
 
 2 
 following : A does not lie between 2 B and 3 B, nor between - B and 
 
NUMBER AND MAGNITUDE. 21 
 
 q 2 3 2 3 
 
 -B, nor between -B and - B, nor between -B and -B, &c. &c. 
 2 ' 3 3 ' 4 4' 
 
 2d, Observe that every affirmative assertion in the above includes a 
 
 certain number of the affirmative ones which precede, and an infinite 
 
 number of parts of the negative ones preceding and following. For 
 
 instance, we find that 100 A lies between 161 B and 162 B, or A lies 
 
 between — B and B, that is between B and 2B. Again, 2 A 
 
 100 100 ' 
 
 322 324 
 
 lies between — B and B, or between 3B and 4B. Similarly, 
 
 100 100 ' ^ 
 
 3A lies between - — B and - — B, or between 4B and 5B; and it 
 100 100 ' 
 
 might thus seem at first as if every affirmation made all the affirm- 
 ations preceding its necessary consequences. But if we try 90 A by 
 the preceding, we shall find that it lies between 
 
 ^^^^^B and il^B or between 144 B and 146 B 
 
 100 100 
 
 so that we can only affirm 90 A to lie either between 144 B and 
 145 B, or between 145 B and 146 B, but we do not (from this) know 
 which. But we can say that 90 A does not lie between 146 B and 
 147 B, or between 147 B and 148 B, &c. The points at which any 
 affirmation does not determine those preceding may be thus found. 
 Let /cA lie between ^B and (Z+1)B; or 
 
 A lies between -B and — r— B 
 mk -T-B and ^ ^ 
 
 If -7^ and — - — lie between t and ^+1, then mX lies between 
 
 ^B and {t +1)B: but if, in going from the first to the second, we pass 
 through a whole number, or iiml, divided by /c, gives a quotient t 
 and remainder r, and m(J,-\-\\ divided by /c, gives a quotient t-{-l 
 and remainder r', then we have 
 
 T "- ^'^l ~r~ ~ ^"^^ "^ k 
 
 or m = k — r -i-r' or r + 171 = k-\-r 
 
 and in all cases where r-{-m is greater than k, this condition can be 
 fulfilled. The process may be shortened, by using instead of / the 
 
22 CONNEXION 01? 
 
 remainder arising from dividing / by k. Suppose, for instance, it is 
 required to determine what preceding affirmatives are ascertained by 
 the proposition 10 A lies between 33 B and 34 B. We have then 
 /=33, kzszlO, remainder of /-r-/cr= 3. 
 
 7W = 2 t= 6 r = 6 2A lies between 6B and 7B 
 
 m=S #= 9 r = 9 ■ 
 
 m = 4 t = l3 7' = 2 4A 13B and 14B 
 
 m = 5 t=l6 r = 5 5A 16B and 17B 
 
 m = 6 t = l9 r = S 
 
 w = 7 jJ=:23 r=l 7A 23B and 24B 
 
 m=zS t=z26 rr=4 
 
 ?w=9 ^ = 29 r-7 
 
 By proceeding thus, it will appear that there is no perceptible law 
 
 regulating the places of A, 2 A, among B, 2B, &c., derivable 
 
 from the sole condition of /c A lying between IB and (/+1) B. Never- 
 tlieless, it is easy to prove, that if all the rest of the relative scale be 
 given from and after any given point, that the whole of the preceding 
 part can then be determined. For, suppose /cB to be the commence- 
 ment of the part of the scale given, and let the place of mA be asked 
 for, which precedes A A, the first multiple of A appearing in the 
 scale. Multiply m byg, so that wg shall be greater than A. Then 
 mgA appears in the portion of the scale given, say between w;B and 
 (w; + l)B. Therefore 
 
 mA lies between -B and B 
 
 S g 
 
 and if — and lie between t and ^ + 1 , the question is settled ; 
 
 o o 
 
 W 
 
 but this must always be the case, if we include the case where - or 
 
 is itself a whole number. 
 
 g 
 From all that precedes, we draw the following conclusions : 
 
 1 . Having given A and B, two incommensurable magnitudes of 
 
 the same species (both lengths, both weights, &c.), we can assign, by 
 
 a processes embling that of finding the greatest common measure in 
 
 arithmetic, the relative scale of multiples of A and B, which points 
 
 out between what two multiples of B any given multiple of A lies, or 
 
 vice versa. 
 
NUMBER AND MAGNITUDE. 23 
 
 2. Any part of the beginning of this scale being deficient, we can 
 construct it by means of the rest. 
 
 3. We can find a magnitude which shall be commensurable with 
 A, differing from B by less than any magnitude we name; and can 
 assign the fraction which it is of A . 
 
 Given the two magnitudes, their relative multiple scale is given ; 
 but when the scale is given, the two magnitudes are not given. For 
 it is easily proved that there is an infinite number of couples of mag- 
 nitudes which have the same scale with any given one. Let the scale 
 
 of A and B be given : then will the scale of- A and - B be the same, 
 
 ^ 9 9 
 
 where p and g are any whole numbers whatsoever. 
 
 For if ^A lie between IB and (I -\- l)B 
 
 then ;fe ^ A lies between Z^B and (Z + 1)^B 
 
 9 9 ^ ^ ' q 
 
 or making ^A = A' ^ B = B' 
 
 9 9 
 
 kA' lies between ZB' and (Z+1)B' 
 
 whence the scale of A and B is the same as that of A' and B' for any 
 value of k. 
 
 What is it, then, which is given when the scale is given ? Not the 
 
 magnitudes themselves ; for if the scale belong to A and B, it also 
 
 P P 
 
 belongs to every one of the infinite cases of -A and -B. The scale, 
 
 therefore, only defines such a relation between the magnitudes as be- 
 longs to 2 A and 2B, 3A and SB, &c., as well as to A and B. It is 
 usual to call this relation the proportion between the two quantities in 
 common life, and in mathematics their ratio ; in Euclid the terra is 
 
 X'oyo?. 
 
 Two magnitudes, A and B, are said to have the same ratio as 
 two other magnitudes, P and Q, when the relative scales of the two 
 are the same ; that is, when the multiples of Q are distributed as to 
 magnitude among those of P, in the same way precisely as those of B 
 are distributed among those of A. And P and Q may be two mag- 
 nitudes of one kind, two areas, for instance, while A and B may be of 
 another, two lines, for instance. 
 
 It is easy to shew that this accordance of scales is equivalent to 
 the common idea of proportion, such as it would become if we took 
 
24 CONNEXION OF 
 
 all means of companson away, except that of multiples. Let us 
 imagine A and B to be two lines in a picture, and P and Q the two 
 corresponding lines in what is meant for an exact copy on a larger 
 scale. Set an artist to determine whether P and Q are in the proper 
 proportion to each other, without any assistance except the means of 
 repeating A, B, P, Q, as many times as he pleases. He will reason 
 as follows : " If Q be ever so little out of proportion to P, though it 
 may not be visible to the eye, yet every multiplication of the two will 
 increase the error, so that at last it will become perceptible. If there 
 be a line 100 A laid down in the first picture, and if it be found to lie 
 between 51 B and 52 B, then should 100 P lie between 51 Q and 
 52 Q. But if Q be a little wrong, then 100 P may not lie between 
 51 Q and 52 Q." 
 
 It only remains to see whether this definition of proportion will 
 include the case of commensurable quantities. These satisfy such an 
 equation as /cA = /B, k and / being two whole numbers, and it is 
 easy to shew that the whole relative scale is divided into an infinite 
 succession of similar portions. Firstly, this one equation determines 
 the whole scale ; for we have 
 
 k k 
 
 or if — lie between ^and^ + 1, mA lies between t B and (^ + 1 )B 
 
 if ^^ = t mA = tB 
 
 7 
 
 Let us suppose, for instance, A = - B. Then we have 
 
 A lies between B and 2B 
 
 2A 3B and 4B 
 
 3A 5B and 6B 
 
 4 A is equal to 7 B : or the scale is 
 
 B A 2B 3B 2A 4B 5B 3A 6B ^^ 
 
 4A 
 From this point the scale begins again in the same order. Thus, 
 the second portion is 
 
 ^^ 8B 6A 9B lOB 6A IIB 12B 7A 13B ^j^ 
 
 4A 8A 
 
 and so on ad infinitum. The arithmetical definition of A having the 
 
NUMBER AND MAGNITUDE. 25 
 
 same ratio to B whicli P has to Q, is simply that of A being the 
 same fraction of B which P is of Q : or if 
 
 A =/b P =/q 
 
 k k 
 
 Now, since the scale depends entirely on r, it is the same for 
 
 both ; conversely, if the scale of A and B be the same as that of 
 P and Q, then if /i;A = /B, /cP must = /Q. Hence the two defi- 
 nitions are synonymous : if one applies, the other does also. 
 
 When the multiple scale of A and B is the same as that of P and 
 Q, we have recognised the proportionality of A and B to P and Q. 
 But these scales may differ. The question now is, may they differ in 
 all possible ways, or how far will their manner of differing in one 
 part of the scale affect their manner of differing in others ? Am I, to 
 take an instance, at liberty to say, that there may be four magnitudes 
 such that 20 A exceeds 18B, while 20 P falls short of 18Q; but that, 
 for the same magnitudes, 13 A falls short of 17B, while 13 P exceeds 
 17 Q ? Such questions as this we proceed to try. 
 
 When only two things are possible, which cannot co-exist, each 
 is the complete and only contradiction of the other : the assertion of 
 one is a denial of the other, and vice versa. But when three different 
 things are possible, one only of which can be true, the assertion of 
 one contradicts both of the other two ; the denial of one does not 
 establish either of the other two. 
 
 The want of a common term, which may simply mean not less, 
 that is, either equal or greater, without specifying which, and so on, 
 causes some confusion in mathematical language. To remind the 
 student that not less does not mean greater, but either equal or 
 greater, we shall put such words in italics. Thus, not less and less, 
 not greater and greater, are complete contradictions : the denial of 
 one is the assertion of the other. 
 
 If A and B be two magnitudes of one kind, and P and Q two 
 others, of the same or another kind, such that 
 
 mA is less than 71 B, ?wP is not less than nQ 
 
 then it is impossible that there should be any multiples such that 
 
 wl'A is greater than n B, Tw'P is not greater than ^'Q 
 
 For we find, from the first of each pair. 
 
26 
 
 CONNEXION OF 
 
 A is less than — B, A is greater than 
 
 B • 
 
 still more is — B greater than — B or — greater than — 
 
 B, A is greater than — ; 
 
 772 ' ° w' 
 
 n 
 
 m - m m 
 
 But P is not less than — Q, P is not greater than — r Q 
 m ° m 
 
 Now, all the four combinations of this latter assertion contradict 
 — is greater than — • ; as follows : 
 
 = — Q, F = — O, gives — = — ; 
 
 P greater than — Q, P = — ^, Q, gives — less than — -. 
 
 m ' m' m m 
 
 P = — Q, P less than — Q, gives — less than — ; 
 
 P greater than — Q, P less than — •, Q, gives — less than —, 
 
 Hence the two suppositions above cannot be true together : the 
 happening of any one case of either proves every case of the other to 
 be impossible. 
 
 If we range all the possible assertions which can be made, we 
 have as follows : 
 
 Wl P is greater than n Q 
 
 mV \^ equal to wQ 
 
 mV is less than wQ 
 
 W«'P is greater than w'Q 
 
 rdV is equal to ?i'Q 
 
 rdV is less than w'Q 
 
 Four of these must be true, one out of each triad ; and there are 
 81 ways of taking one of each, so as to put four together. But we 
 shall take the sets A and a together, and find what inference we can 
 draw by taking one out of each. 
 
 As 
 
 mA is greater than ?^B 
 
 P3 
 
 A. 
 
 7W A is equal to nB 
 
 p. 
 
 Ai 
 
 mk is less than w B 
 
 Pi 
 
 «3 
 
 mPi. is greater than ?^'B 
 
 pt 
 
 ^2 
 
 m'A. is equal to w'B 
 
 Vi 
 
 H 
 
 niK is less than w'B 
 
 Ih 
 
 A3 ^3 proves nothing as to — and —7. It merely says that 
 
 — B and — ; B are both exceeded by A, which may be whether 
 m m 
 
 — is greater than, equal to, or less than —,. The same for P3 p^ 
 
NUMBER AND MAGNITUDE. 
 
 27 
 
 A3 a^ proves — , greater than — ; as does P3 p^ 
 
 A3 tti proves —7 greater than — ; as does P3 p^ 
 
 A2 ^3 proves —7 less than — ; as does Pg p^ 
 A2 «2 proves — -, equal to — ; as does Pg pg 
 
 A2 «! proves —7 greater than — ; as does Pg p^ 
 
 Ai a^ proves —7 less than — ; as does P^ p 
 
 ; as does P^ p^ 
 neither does Pi Pi 
 
 Ai «2 proves — , less than 
 
 Ai tti proves nothing ; 
 
 Now, if we put these pairs together, or make pairs of assertions, 
 in the manner already done, we have 81 distinct sets of four asser- 
 tions, divisible into those which mat/ be true together, and those which 
 cannot be true together. An inconsequential supposition, such as 
 Agflg, may co-exist with any of the rest from the other set Vpi but 
 
 those which give —7- necessarily greater, equal to, or less than — in 
 
 the set A a, can only co-exist either with the similar ones from the 
 set P/), or with those which are inconsequential. Thus we have 
 
 Vp 
 
 A3 053 may be true with any marked 
 A3 Cf2 requires either P3J53, 
 
 A3fli 
 A2G3 
 
 •"2 ^2 
 
 Agfli 
 
 Ai«3 
 
 A^a 2 
 
 P3P3, 
 
 Ps/^s, 
 PsPs, 
 Psi^s. 
 
 P3P3, 
 
 Vzpl, 
 
 Psi'i, 
 
 p^p.. 
 
 Pai'i, 
 
 P2P3, 
 
 P.?3, 
 
 Pai'j. 
 
 
 PsPa. 
 
 Psi'i. 
 
 P2P3. 
 
 P.i's, 
 
 'PiPz, 
 
 Pli>3, 
 
 P2P1, or Pi;?! 
 
 P2;>i, •• Pii?i 
 
 Pii?2, •• Pii^i 
 
 •• Pi/^i 
 
 P2P1, •• Pi^i 
 
 Pi;?2, •• PlJOl 
 
 ^lP2> " Pli^l 
 
 Ai «! may be true with any marked Pp 
 
 The remaining thirty cannot be true ; but it is unnecessary to specify 
 
28 CONNEXION OF 
 
 them, as a simple induction from the preceding will shew how to 
 classify those which may and cannot be true. Attach an idea of 
 magnitude to the phrases greater ^ equal, and less; say that " A is 
 greater than Bj" is higher than " A is equal to B," and this again 
 higher than " A is less than B" We have marked the highest 
 phrases by the highest numbers. Say that in AgOj, A3 «,, &c. (calling 
 A and a the antecedent clauses of any four marked A, a, P, p), the 
 antecedents are descending; in AgWg, Aga^, and A, a,, stationary ; 
 and in Aj Oj* Aj a^, &c. ascending. Then all the propositions which 
 imply the co-existence of any two antecedents, and any two conse- 
 quents of the form A a Pp,-may be divided into those which may be 
 true, and those which cannot be true, by the two following rules : 
 Ascending antecedents cannot have descending consequents. 
 Descending antecedents cannot have ascending consequents. 
 
 Precisely the same rules will apply if we take two propositions 
 A P for antecedents, and two others ap for consequents; as we may 
 either deduce in the same manner, or by simple inversion. For if 
 A« Pp, with any numerals subscribed, do not contradict either of the 
 preceding rules, neither will the corresponding case of A P aj5 do so, 
 and the contrary. Instances, Aj a^ P^pg and A, P3 a^p^ ; k^a^ ^zVi 
 and A2P3 a^p^y &c. 
 
 Let us then take a case of A, B, P, Q, in which we find one 
 ascending assertion relative to mA, wB, twP, nQ, for some particular 
 values of m and n ; for instance 
 
 * p rSA is less than 4B 
 ^ ^ tsP is greater than 4Q 
 
 which, as we have seen, is never contradicted in form by any assertion 
 that can be true of any other multiples. These four quantities are 
 not proportionals: for 3 A being less than 4B, and 3P greater than 
 4Q, P cannot lie in the scale of P and Q in the same place as A in the 
 scale of A and B. But to what more common notion can we assi- 
 milate this sort of relation between A, B, P, and Q, namely, that 
 all true assertions of the form (AP) are either ascending or stationary, 
 and never descending ? Have we any thing corresponding to this in 
 the arithmetic of commensurable quantities? Let us suppose A and 
 B commensurable, and also P and Q : say that 
 
 
NUMBER AND MAGNITUDE. 29 
 
 t t' ■ 
 
 Then 3-B is less than 4 B ; 3-Q is greater than 4Q; 
 
 - IS less than - ; -, is greater than - : -, is greater than - 
 V 3 V ^ 3 u V 
 
 or A is a less fraction of B than P is ofQ; which in arithmetic is 
 
 also said thus, A bears a less proportion to B than P does to Q, or 
 
 P bears to Q a greater proportion than A bears to B. Hence we get 
 
 the following definitions, in which we insert the previous definition of 
 
 proportion, and the accordance of the whole will be seen. 
 
 When all true assertions on (mA, wB) (mP, rzQ) are either 
 ascending or stationary, and never descending, A is said to have to B 
 a less ratio than P to Q; when always stationary, the same ratio; 
 when always descending or stationary, and never ascending, a greater 
 ratio. 
 
 This amounts in fact to the definition given by Euclid, the 
 opening part of whose Fifth Book we shall now make some extracts 
 from, with a few remarks. 
 
 Definition III. Ratio is a certain mutual habitude (jr^icis, 
 method of holding or having, mode or kind of existence) of two 
 magnitudes of the same kind, depending upon their quantuplicity 
 (^^tikixorvt, for which there is no English word ; it means relative 
 greatness, and is the substantive which refers to the number of times 
 or parts of times one is in the other). 
 
 In this definition, Euclid gives that sort of inexact notion of ratio 
 which defines it in commensurable quantities, and gives some light 
 as to its general meaning. It stands here like the definition of a 
 straight line, "that which lies evenly between its extreme points" 
 prior to the common notion, " two straight lines cannot enclose 
 space," which is the actual subsequent test of straightness. In most 
 of the editions of Euclid we see " Ratio is a mutual habitude of two 
 magnitudes with respect to quantityy^ which makes the definition 
 unmeaning. For quantity and magnitude in our language are very 
 nearly, if not quite, synonymous; or if any distinction can be drawn, 
 it is this : magnitude is the quantity of space in any part of space. 
 But as Euclid is here speaking of magnitude generally (not of space 
 magnitudes only) the words magnitude and quantity are the same.* 
 
 * Euclid again uses the word -rvXixorr,; (Book VI. def. 5) in a 
 manner which settles its meaning conclusively. The more advanced 
 reader may consult Wallis, Opera Mathematica, v. 11. p. 665. 
 
 D 2 
 
30 CONNEXION OF 
 
 Definition IV. Magnitudes are said to have a ratio to each 
 other which can, being multiplied, exceed " one the other." This 
 means that quantities have a ratio when, any multiples of both being 
 taken, the relation of greater or less exists. It is usually rendered 
 " Two magnitudes are said to have a ratio when the lesser can be 
 multiplied so as to exceed the greater." But the above is literally 
 translated, and the sense here given to ratio makes the next definition 
 consistent. It is a way of expressing that the two magnitudes must 
 be of the same kind, which requires that the notion of greater and less 
 should be applicable to them. That this notion should be applicable 
 to the quantities themselves as well as their multiples, being the 
 necessary and sufficient condition of the possibility of the comparison 
 implied in the next definition, is here assumed * as the distinction of 
 quantities which have a ratio. 
 
 Definition V. Magnitudes are said to be in the same ratio 
 the first to the second, and the third to the fourth : when the same 
 multiples of the first and third being taken, and also of the second 
 and fourth, with any multiplication, the first and third (multiples) are 
 greater than the second and fourth together, or equal to them together, 
 or less than them together. 
 
 This amounts to our definition of proportion, namely, that the 
 relative multiple scale of A and B is the same as that of P and Q. 
 For, take the same multiples of A and P, namely, mA and mP, and 
 the same multiples of B and Q, namely, wB and wQ. Then, if the 
 relative multiple scales be the same, let wA lie between vB and 
 (u-|-l)B, it follows that 7nF lies between vQ and (t;-{-l)Q. If, then, 
 w be less than v,nB is less than vB, and nQ less than vQ. And 
 772 A being greater than vB must be greater than nB, while, for the 
 same reason, mP is simultaneously greater than nQ. In the same 
 way the other parts of the definition V. may be shewn to be included 
 in that of identity of multiple scales. Now, reverse the supposition 
 
 * The common version is several times referred to afterwards, and the 
 definition 4 expressly alluded to, in the editions of Euclid. But it must 
 be remembered that the Greek of Euclid contains no references to pre- 
 ceding propositions, these having been supplied hy commentators. The 
 reader may, if he can, make Aoyov s^^stv -r^os Sckktikcc /xiyi^tj kiysrai, 
 a ^vvarai ToXXx'vXaffta.^ofAiva. aXXrikcov ifri^ix,iiv mean, " Magnitudes are 
 said to have a ratio, when the less can be multiplied so as to exceed 
 the greater." 
 
NUMBER AND MAGNITUDE. 31 
 
 and assume Euclid's definition. If, then, mA lie between vB and 
 (v+l)B, it follows that mA is greater than vB, whence, by the 
 assumption mP is greater than vQ. Similarly, because w A is less 
 than (v+l)B, TwP is (by that definition) less than (■y+l)Q. There- 
 fore, mV lies between vQ and (vH-l)Q, or in this instance, or for 
 any one value of w, the scales are accordant, and the same may be 
 proved in any other case. It follows, then, that the two definitions 
 are mutually inclusive of each other. 
 
 The manner in which Euclid arrived at this definition has been 
 matter of inquiry. But any one who will examine the first nine 
 propositions of the tenth * book, will see that he had precisely the 
 same means of arriving at it as we have used. But, besides this, he 
 might have come by the definition from a common notion of practical 
 mensuration, as follows. Suppose two rods given, one of which is 
 the English yard, the other the French metre, but neither of them 
 subdivided. The only indication which looking at them will offer, is 
 that the metre exceeds the yard apparently by about ten per cent. 
 To get a more exact notion, the obvious plan will be to measure some 
 great distance with both. Suppose 100 yards to be taken off with 
 the yard measure, it will be found that that 100 yards contains about 
 91 metres and a half, the half being taken by estimation, and we will 
 suppose the eye could not thus err by a quarter of a metre. Then 
 the yard must be -915 nearly of a metre, and the error upon one yard 
 cannot exceed the hundredth part of the quarter of a metre, or '0025 
 of the metre. But the mathematician, to make this process perfectly 
 correct, will suppose distance ad infinitum, measured from a point 
 both in yards and metres, or in fact will form what we call the relative 
 multiple scale. He then looks along this scale for a point at which 
 a multiple of a yard, and a multiple of a metre end together. If 
 this happen, and it thus appear that m yards is exactly equal to 
 
 n 
 n metres, the question is settled, for a yard must be — of a metre. 
 
 But it will immediately suggest itself to a mind which is accustomed 
 not to receive assumptions without inquiry, that it may be no two 
 
 • There are two English editions of the lohote of Euclid, and there 
 may be more : that of John Dee (now old and very scarce) and that of 
 J. Williamson, London, 1788, in two thin quarto volumes. The disser- 
 tations in the latter are a strange mixture of good and bad, but the text 
 is very literally Euclid, in general. 
 
32 CONNEXION OF 
 
 points ever coincide on the multiple scale. But in this case it is 
 very soon proved tliat mA may be made as nearly equal to nB as 
 
 we please, by properly finding m and n; so that a fraction — may 
 
 be found such that A shall be as nearly — B as we please. Even 
 
 m ^ 
 
 admitting that this would do to assign A in terms of B, it leaves us 
 no method of establishing any definite connexion between A con- 
 sidered as a part of B, and P considered as a part of Q. 
 
 The word part usually means arithmetical part, namely, the 
 
 3 
 result of division into equal parts. Thus - is a part of 1 made by 
 
 dividing 1 inio 7 equal parts, and taking 3 of them. The phrase of 
 
 Euclid in the books on number (VII. to X. both inclusive) is that - is 
 
 3 
 part of 1, - h parts of 1. And it is easily shewn that, in this use of 
 
 the word, every quantity is e.\\hQx part ox parts of every other quantity 
 which is commensurable with it. And of two incommensurable quan- 
 tities, neither is part or parts of the other. But in the original sense 
 of the word part, any less is always part of the greater. This notion 
 of incommensurability, the non-existence of the equation m A = wB, 
 for any values of m or n, obliges us to have recourse to a negative 
 definition of proportionality, a term which we proceed to explain. 
 Examine the definition of a square, namely, " a plane foursided figure, 
 with four equal sides and one right angle." It is clear that the ex- 
 amination of a finite number of questions will settle whether or no a 
 figure is a square. Has it four sides ? are they in the same plane? 
 are the sides equal ? is one angle a right angle? Froof of the affirm- 
 ative of these four propositions proves the figure to be a square. Now, 
 examine the number of ways in which a figure can be shewn to be 
 not a square. All propositions are either affirmative or negative ; 
 A is B or A is not B. The affirmative can be proved or the negative 
 disproved, with one result only, for both give A is B. But the 
 affirmative can be disproved, or the negative proved, with an infinite 
 number of results ; it is done by proving that A is C, or D, or E, &c. 
 &c. ad infinitum. Thus there may be an infinite number of ways of 
 shewing that a figure is not a square, but there is only one way of 
 shewing that it is a square. This we call di positive definition. 
 
 Now examine the definition of parallel lines, " those which are in 
 
NUMBER AND MAGNITUDE. 33 
 
 the same plane, but being produced ever so far do not meet." We 
 are not considering where the lines meet, if they do meet, or dis- 
 tinguishing between lines which meet in one point and in another, but 
 simply dividing all possible pairs of lines into two classes, parallels 
 and intersectors. Now here it is impossible to prove * the affirmative 
 of the proposition, " A and B are parallels," by means of the de- 
 finition only, without proving an infinite number of cases. To see 
 this more clearly, remember that every proposition relative to the 
 intersection or non-intersection of straight lines, is an assertion which 
 either includes or excludes every possible couple of points which can 
 be taken, one on each straight line. " Lines intersect" means there 
 is a couple of such points which coincide. " Lines are parallel " 
 means that tliere is no such couple whatsoever, of all the infinite 
 number which can be taken. 
 
 The first proposition in which Euclid proves the existence of 
 parallels (the 27th) does not shew that the lines are parallels, but 
 that the proposition, " the lines are intersectors," is inconsistent with 
 preceding results. The proposition, " A and B are parallels," though 
 it appears affirmative, yet is in Euclid a negative, for his express 
 definition of parallels does not define what they are, but what they are 
 not, " not intersectors." This we call a negative definition. 
 
 Now, to examine further Euclid's definition of equal ratios, we 
 must consider his definition of greater and less ratios. They amount 
 to the following. A is said to have to B a greater ratio than P has 
 to Q where there is, among all possible whole numbers m and w, 
 ani/ one pair which give niA greater than nB, but mP equal to or 
 less than wQ; or which give mA equal to nB, but mF less than nQ : 
 which give in fact, in any one case, what we have called a descending 
 assertion. And A is said to have to B a less ratio than P has to Q, 
 when any one pair of whole numbers m and n gives mA less than 
 TwB, but mP equal to or greater than wQ, or mA equal to nB, but 
 otP greater than wQ : which give in fact, in any one case, what we 
 have called an ascending assertion. Here, to a mind the least inqui- 
 sitive, appears at once a decided objection. Our notions of the terms 
 
 * The celebrated axiom of Euclid evades this, and in point of fact 
 amounts to another and a positive definition of parallels, the assumption 
 being that the old definition agrees with it. Or rather we should say, 
 that the first twenty-five propositions of the first book establish a part of 
 the connexion of the definitions, and the axiom assumes the rest. 
 
34 CONNEXION OF 
 
 greater and less will never allow us to suppose that any thing, quan- 
 tity, ratio, or any thing else, can be both greater and less than another 
 quantity, or ratio ; and yet, on looking at the definition of Euclid, 
 we see that for any thing which appears to the contrary, one pair of 
 values of m and n may shew that A has a greater ratio to B than P 
 to Q, while another pair may shew that it has a less. The objection 
 is perfectly valid ; the only fault to be found is, that it should not 
 have arisen before, when the definitions of the first book were pro- 
 posed. How is it then known that there can be such a thing as a 
 foursided figure with equal sides and one right angle, or as lines 
 which never meet ? The confusion arises from placing the definitions 
 in the form of assertions, before the possibility of the assertions which 
 they imply are proved. The defect may be remedied (we take the 
 square as an instance) in two ways. 
 
 1. Write all definitions in the following manner. To define a 
 square, for example, " if it be possible to construct a plane figure 
 having four equal sides and one right angle, let that figure be called 
 a square." 
 
 2. Omit the definition of a square, head the 46th proposition of 
 the first book as follows. 
 
 " Theorem. On a given straight line, a four-sided figure can be 
 constructed which shall have all its sides equal to the given straight 
 line, and all its angles right angles." Having demonstrated this, add 
 the following definition : Let the figure so constructed be called 
 a square. 
 
 We have shewn that all sets of four magnitudes, A and B of 
 one kind, P and Q both of the same kind with the first, or both of 
 one other kind, can be divided into three classes. 
 
 1. Those in which simultaneous assertions on mA and n B, and 
 on mP and nQ, are all (for all values of m and n) either ascending 
 or stationary. 
 
 2. Those in which they are all stationary. 
 
 3. Those in which they are all either descending or stationary. 
 For we have shewn that the only remaining possible case a priori, 
 
 namely, that in which there are both ascending and descending 
 assertions for different values of m and w, is a contradiction amounting 
 in iact to supposing one fraction to be both greater and less than 
 another. And it has been shewn that all the three cases are possible, 
 for commensurable quantities at least. We are now, therefore, in a 
 
NUMBER AND MAGNITUDE. 35 
 
 condition to say, let A and B in the first case be said to have a 
 less ratio to B than P has to Q; in the second, the same ratio; in 
 the third, a less ratio. The only question now is, are tliese definitions 
 properly negative or positive. It will immediately appear that, out 
 of the three, the first and third can be directly and affirmatively 
 shewn to be true of particular magnitudes, and that the second cannot. 
 By which is meant, that the comparison of individual multiples may, 
 by a single instance, establish the first or third, but that no com- 
 parison of individual multiples, however extensive, can establish the 
 second. For the second consists in stationary assertions ad irifinitumj 
 and the first and third are proved by a single ascending or descending 
 assertion. 
 
 As an instance, suppose 
 
 A = 951 feet B = 497 feet 
 1902 994 
 2853 1491 
 3804 1988 
 4755 2485 
 
 P = 1300 lbs. Q= 679 lbs. 
 2600 1358 
 3900 2037 
 5200 2716 
 6500 3395 
 
 In these first five multiples, there are none but stationary assertions, 
 of twenty five which might be made. Thus 
 
 4755 > 994 1 2853 > 2485 1 951< 994 j ^^ 
 6500 >1 358 j 3900 > 3395 J 1300 < 1358] 
 
 but neither of the three definitions is thereby shewn to belong to these 
 four magnitudes. Now, take the first and third 498 times, and the 
 second and fourth 952 times, and we have, going on with the series 
 of multiples, 
 
 473598 473144 647400 646408 
 474549 473641 648700 647087 
 
 and here the process may close, for we have 473598 less than 473641, 
 while 647400 is greater than 647087. Consequently, we have 
 proved, by comparison, that 951 feet has to 497 feet a less ratio than 
 1300 lbs. to 679 lbs. 
 
 But the case in which neither greater nor less ratio exists can never 
 be established by actual comparison of multiples, except only in the 
 case where the pairs of magnitudes are commensurable. For, remark 
 that the mere circumstance of the relative multiple scale of A and B 
 
36 CONNEXION OF 
 
 agreeing with that of P and Q up to any point, is neither proof nor 
 presumption that the two magnitudes given are actually proportional, 
 though, as we shall see, it is certain evidence that they are nearly 
 proportional, if the multiple scales agree for a great number of 
 multiples. Proportion is not established until the similarity of the 
 multiple scales is shewn to continue for ever. Now, though it would 
 not be remarked at first, this insertion of an infinite number of con- 
 ditions to be fulfilled, is tantamount to a negative definition, if we 
 wish to make the definition specifically speak of one absolute cri- 
 terion of disproportion or proportion. Disproportion is where there 
 is an ascending or descending assertion somewhere in the comparison 
 of the multiple scales. Proportion is where there is no descending 
 or ascending assertion. 
 
 In the case of commensurable quantities the definition is positive, 
 because there is then a single stationary assertion, which, being proved, 
 all the rest are shewn to follow. If A and B be commensurable, let 
 w A = w B ; then if w P = n Q, there is proportion ; if not, there is 
 disproportion. See page 24 for the proof as to the rest of the mul- 
 tiple scales. 
 
 We have said, that, when the multiple scales agree for a long 
 period, there is proportion nearly ; and it is proved thus : Suppose 
 that the scale of A and B agrees with that of P and Q, up to 10,000 
 P and 10,000 Q, but that we have disagreement as follows: 9326 A 
 lies between 10,000 B and 10,001 B, whereas 9326 P lies between 
 10,001 Q and 10,002 Q. Or the scales run thus : 
 
 10,000 B 9326 A 10,001 B 10,002 B 
 10,000 Q 10,001 Q 9326 B 10,002 Q 
 
 How much must we alter A to produce absolute proportion ? Not 
 more than would be necessary to make 9326 A greater than 10,002 B, 
 or less than would still keep it less than 10,001 B. That is, we must 
 so alter A as to add somewhere between and 2B to 9326 A, or 
 somewhere between 
 
 and r-^B to A 
 
 9^26 
 
 Consequently, the addition of a small part of B to A would make an 
 accurate proportion. 
 
 We might now proceed to the propositions of the Fifth Book of 
 
NUMBER AND MAGNITUDE. 37 
 
 Euclid ; but there are three difficulties in the way of the student's 
 perfect satisfaction with the definition. 1st, He may have a mys- 
 terious idea of incoramensurables. 2d, He may not be satisfied of 
 the necessity of departing from arithmetic. 3d, He may find it diffi- 
 cult to imagine how the existence of proportionals can ever be esta- 
 blished, with, apparently, an infinite number of conditions of definition 
 to satisfy. We suppose that the gravity of tone which elementary 
 writers adopt, is inconsistent with the statement of a beginner's diffi- 
 culties, in the words in which he would express them. We shall 
 remove all necessity for preserving such dignity in a case where it 
 may be inconvenient, by a simple supposition. Let -4 be a beginner 
 in the stricter parts of mathematics ; that is, a person apt to mix pre- 
 viously acquired notions with the meaning he attaches to definitions 
 which are intended to exclude all but the ideas literally conveyed in 
 the words which are used; much better pleased with the apparent 
 simplicity of an incorrect definition, gained either by omitting what 
 should not be omitted, or by supposing what cannot be supposed, 
 than with the comparatively cumbrous forms which provide for all 
 cases, and distinguish differences which really exist; and, finally, 
 when a doubt exists, rather predisposed against, than in favour of, the 
 necessity of demonstration. Let B be another person, who has sub- 
 jected his mind to that sort of discipline which has a tendency to 
 remove the propensities abovementioned. We can imagine them 
 talking together in this manner : 
 
 A. — I have been trying to understand the meaning of incommen- 
 surable quantities, and cannot at all make out how it can be that one 
 given line may be no fraction whatsoever of another given line, though 
 both remain fixed, and certain lines ever so little greater or less than 
 the first are fractions of the second. 
 
 B. — A little consideration will teach you, that neither in arithmetic 
 nor geometry are we at all concerned with how things can be, but only 
 with whether they are or not. Do you admit it to be demonstrated 
 that the side and diagonal of a square, for instance, are incom- 
 mensurable ? {Algebra, page 98). 
 
 A. — I cannot deny the demonstration, but the result is incompre- 
 hensible. Does it really prove, that if I were to cut the diagonal of a 
 square into ten equal parts, each of these again into ten equal parts, 
 and so on for ever, I should never, by any number of subdivisions, 
 
 £ 
 
38 CONNEXION OF 
 
 succeed in placing a point of subdivision exactly upon the point 
 which cuts oflf a length equal to the side. 
 
 JB. — I take it for granted you have sufficiently comprehended the 
 definitions of geometry, to be aware that a thin rod of black lead, or 
 a canal of ink, are not geometrical lines ; and that the excavations 
 which you perforate by the compasses are not points. 
 
 A. — Certainly ; I now have no difficulty in imagining mere length 
 intersected by partition marks, which are not themselves lengths. 
 
 B. — Then, in the case you proposed, you need not go so far for 
 a difficulty ; for your method of subdivision will never succeed in 
 cutting off so simple a fraction as the third part of the diagonal. 
 
 J.— Why not? 
 
 B. — You see that 9, 99, 999, &c., are all divisible by 3, so that 
 10, 100, 1000, &c., cannot in any case be divisible by 3, but must 
 leave a remainder. Your method of subdivision can never put to- 
 gether any thing but tenths, hundredths, &cc. If possible, suppose 
 one-third to be made up of tenths, a in number, added to hundredths, 
 b in number, added to thousandths, c in number. Then we must have 
 
 -=- + -^ +-^ 
 3 10 ^ 100 ^ 1000 
 
 Clear the second side of fractions, and we have 
 
 i^ = «x 100 + Z»x 10 + c 
 
 or is a whole number, which is not true. And the same rea- 
 
 3 ' 
 
 soning might be applied to any other case. 
 
 A. — This is conclusive enough ; but it seems to follow that the 
 third part of a line is incommensurable with the whole. 
 
 B. — So it is, as far as the one method of subdividing which you 
 propose is concerned. Let tenths, hundredths, &c., be the measurers, 
 and one-third and unity are incommensurable. But the word with 
 which we set out implies all the possible subdivisions of halves, thirds, 
 fourths, fifths, &c. &c., to be tried, and all to fail. 
 
 A. — But here is an infinite number of ways of subdividing. Can 
 it be possible that no one of them will give a side of a square, when 
 the di igonal is a unit ? 
 
 B. — In the first place, it would be a sufficient answer to this sort 
 of difficulty to say, that, for any thing you know to the contrary, the 
 
NUMBER AND MAGNITUDE. tJ9 
 
 number of ways in which you may fail is as infinite as the number of 
 ways in which you may try to succeed. In the second place, there 
 is also an infinite number of ways of subdividing, which will not give 
 one-third* Let your first subdivision be into any number of equal 
 parts, except only 3, 6, 9, 12, &c.; and your second subdivision the 
 same, or any other, with the same exceptions, &c. The same rea- 
 soning will prove that you can never get one-third. 
 
 A. — But look at the matter in this way. Suppose the halves, the 
 thirds, the fourths,' the fifths, 8cc. &c. of a diagonal laid down upon it 
 nd infinitum, so that there is no method of subdividing into aliquot 
 parts, how many soever, but what is done and finished. Would not 
 the whole line be then absolutely filled with subdivision points, and 
 would not one of them cut off a line equal to the side of the square. 
 
 B. — You have now changed your use of the word infinite, and 
 applied it in the sense of infinity attained, not infinity unattainable. 
 As long as you used the word to signify succession, which might be 
 carried as far as you pleased, and of which you were not obliged to 
 make an end, the word was rational enough, though likely to be mis- 
 understood ; but as it is, you may as well suppose you have got be- 
 yond infinite space, at the rate of four miles an hour, and are looking 
 back upon the infinite time which it took you to do it, as imagine 
 that you have subdivided a line ad infinitum. But if the idea of in- 
 finity attained be a definite conception of your mind, you meet the 
 difficulty of incommensurable quantities in another form. The defi- 
 nition of the term incommensurable was shaped in accordance with 
 the exact notion, that, subdivide a line as far as you may, you must 
 stop at some finite subdivision ; and incommensurable parts of a 
 whole are those which you never exactly separate arithmetically, stop 
 at what finite subdivision you please. But, if you will contend for 
 infinite subdivision attained, and imagine the line thus filled up by 
 points, then it will be necessary to divide all parts of a whole into two 
 classes, those which are cut off by finite subdivision, and those which 
 are not attainable, except by infinite subdivision ; the former answering 
 to commensurable, the latter to incommensurable, parts. The diffi- 
 culty remains then just as before ', in other words, why should the 
 side of a square be not attainable from its diagonal except by infinite 
 subdivision, when the sides of a rectangle, which are as 3 to 4 (instead 
 of 3 to 3), are attainable by a finite number of subdivisions ? 
 
 In the next place, you have spoken of a line filled up by points, 
 
40 CONNEXION OF 
 
 the infinitude of the number of points being the compensation for each 
 of the points having no length whatsoever ; at least, it is not easy to 
 see what else you can mean. 
 
 A. — Certainly that is what I mean ; and the common expressions 
 of algebra are in accordance with what I say. For, if I cut aline 
 into n equal parts, it is plain that the sum of the n parts makes up 
 the whole, be the number n great or small. But by making n suflS- 
 ciently great, each of the parts may be made as small as I please ; 
 and, therefore, allowing it to be rational to say that P takes place 
 when n is infinite, in all cases in which we may come as near to P as 
 we please, by making n. sufficiently great (which is the expressed 
 meaning of infinite in algebra), it follows that we may say, that the 
 line is made up of the infinite number of points into which it is cut 
 when divided into an infinite number of equal parts. 
 
 JB. — I see every thing but the last consequence. 
 
 A. — Why, surely, the smaller a line grows, the more nearly does 
 it approximate to a point. 
 
 B. — How is that proved ? 
 
 A. — Suppose two points to approach each other, they continually 
 inclose a length which is less and less, and finally vanishes altogether 
 when the two points come to coincide in one point. So that the 
 smaller the straight line is, the more near is it to its final state — a 
 point. 
 
 JB. — You have not kept strictly to your own idea (which is a 
 correct one) of the way in which the words nothing and infinite may 
 be legitimately used. You have supposed a line to be entirely made 
 up of points, each of which has no length whatsoever, because you 
 may compose a line of a very large number of very small lines, each 
 of which, you say, is nearly a point. Let us now consider whether 
 your final supposition is one to which we can approach as near as we 
 please by diminution of a length. Any line, however small, can be 
 divided into other lines by an infinite number of different points ; for 
 any line, however small, admits of its halves, its thirds, &c. &c. So 
 that there is a theorem which is not lessened in the numbers it speaks 
 of, or altered in force or meaning, in any the smallest degree, by 
 diminishing the line supposed in it ; namely, any line whatsoever 
 admits of as many different points as we please being laid down in it. 
 "Now f of yom final length, or limit of length — the point — this is not 
 true : consequently, you throw away a result at the end, which you 
 
NUMBER AND MAGNITUDE. 
 
 41 
 
 cannot throw away as nearly as you please during the process by 
 which you attain that end ; nor will the denial of it, near the end, be 
 less in the consequence or amount of the error, than if the rejection 
 were made further from the end. Therefore, in asserting that a dimi- 
 nishing straight line approximates to a point, you have abandoned 
 the condition under which you are allowed to speak of nothing or 
 infinite. 
 
 Again, the wth part of a line taken twice is certainly greater than 
 the simple nth part, however great n may be. Now, what do you 
 suppose two points to be, which are laid side by side without any in- 
 terval of length between them ? 
 
 A. — They are, of course, one and the same point. 
 
 JB. — But in your infinite subdivision, two nth parts must be greater 
 than one nth part, or two of your points must be greater than one ; 
 but these two points are the same point, which is therefore twice as 
 great as itself. Such are the consequences to which the supposition 
 of a line made up of points will lead. 
 
 A. — I have frequently heard of lines being divided into an infinite 
 number of equal parts. 
 
 B. — But you never heard those equal parts called points. I 
 can soon shew you that, in the mode of allowing infinity to be 
 spoken of, this fundamental condition is preserved, namely, that no 
 theorem, limitation, number, nor other idea whatsoever, which forms 
 a part of any question, is allowed to be rejected or modified when n is 
 infinite, unless it can be shewn that such rejection or modification 
 may be made with little error when n is great, with less error when n 
 is greater, and so on ; finally, with as small an error as we please, by 
 making n sufficiently great. Now, remark the following truths, and 
 the form of speech which accompanies them, when n is supposed 
 infinite. 
 
 General Theorem. 
 The greater the number of 
 equal parts into which a line is 
 divided, the less line is each of 
 the parts : so that an aliquot 
 part of any line, however great, 
 may be made less than any 
 given line, however small. 
 
 Terminal Theorem. 
 If a straight line be divided 
 into an infinite number of equal 
 parts, each part is an infinitely 
 small line. 
 
 E 2 
 
42 
 
 CONNEXION OP 
 
 General Theorem. 
 
 Any line, however small, 
 may be cut by as many points 
 as we please. 
 
 No straight line, however 
 small, ceases to be a length ter- 
 minated by points. 
 
 Terminal Theorem. 
 
 An infinitely small line may 
 be cut by as many points as we 
 please. 
 
 An infinitely small straight 
 line is a length terminated by 
 points. 
 
 Now, taking your notion of infinite subdivision attained, it may 
 be shewn that incommensurable parts necessarily follow. For, how- 
 ever far you carry the subdivision, you do not, by means of the sub- 
 division points, lessen the number of points which may be laid down. 
 For each interval defined by the subdivisions contains an infinite 
 number of points. Consequently, if you will suppose the infinite 
 subdivision attained, you cannot do it without supposing an infinite 
 number of points left in the intervals, or an infinite number of in- 
 commensurable quantities. This I intend only to shew that the 
 proof of the existence of incommensurable quantities is, upon your 
 own supposition, somewhat better than that of their non-existence. 
 But it would be better to use nothing and infinity as convenient 
 phrases of abbreviation, not as containing definite conceptions which 
 may be employed in demonstration. 
 
 A. — I do not see how your objection applies against nothing; 
 if we cannot attain infinity by continual augmentation, we can cer- 
 tainly attain nothing by continual diminution. 
 
 B. — So it may seem at first, and in truth you are right as to one 
 sort of diminution, that which is implied in the word subtraction. 
 From the place in which there is something take away all there is, 
 and you get nothing by a legitimate process. But subtraction is the 
 only process which leaves nothing ; division, for example, never leaves 
 it. Halve a quantity, take the half of the half, and so on, ad infinitum: 
 you will never reduce the result to nothing. 
 
 A. — But however clearly you may shew that incommensurable 
 quantities actually exist, as a necessary consequence of our de- 
 finitions of length, number, &c., I should feel better satisfied if 
 you could give something like an account of the way in which 
 they arise. 
 
 B. — If you will consider the way in which number and length 
 are conceived, perhaps the difficulty may be somewhat lessened. Let 
 
NUMBER AND MAGNITUDE. 43 
 
 a point set out from another point, and move uniformly along a 
 straight line until the two are a foot distant from each other. It is 
 clear that every possible length between and one foot will have 
 been in existence at some part or other of the motion. Now, suppose 
 a number of points as great as you please, to set off from the first 
 point together ; but, instead of moving in the straight line let them 
 
 move off in curves, the first coming to the straight line at - and 1 
 
 12 12 3 
 
 of a foot : the second at - - and 1 ; the third at - - - and 1 of a 
 ' 3 3 4 4 4 
 
 foot ; and so on, as in this diagram. 
 
 Can you feel sure that these contacts of curves with the line, 
 separated as they must always be from each other by finite intervals, 
 will ever fill up the whole line described by a continuous motion. 
 If not, this figure will always supply presumption in favour of in- 
 commensurable parts, which will of course be increased to certainty 
 by the actual proof of their existence. And this should be sufficient 
 to overturn a doubt which after all is derived from confounding the 
 mathematical point with the excavation made by the points of a pair 
 of compasses. The practical commensurability of all parts with the 
 whole is a consequence of there being magnitudes of all sorts below 
 the limits of perception of the senses (see page 3). 
 
 A, — Granting, then, that there are such things as incommensurable 
 quantities, it is admitted, that though A and B are incommensurable, 
 yet A and B -|- K may be made commensurable, though it be insisted 
 on that K shall be less than any given quantity, say less than the 
 liundred thousand million millionth of the smallest quantity which 
 the senses could perceive, if they were a hundred thousand million 
 of million of times keener than they are at present. Would it not be 
 sufficient, when incommensurable quantities, A and B, occur, to 
 suppose so slight an alteration made in B as is implied in the above, 
 and reason upon A and B + K so obtained, instead of upon A and B. 
 Surely such a change could never produce any error which would be 
 of any consequence ? 
 
 B. — Of consequence to what? 
 
 k 
 
44 CONNEXION OF 
 
 A. — To any purpose of life for which mathematics can be made 
 useful. 
 
 B. — I am still at a loss. 
 
 A. — What process in astronomy, optics, mechanics, engineering, 
 manufactures, or any other part either of physics or the arts of life, 
 would be vitiated by such an alteration, or its consequences, to any 
 extent which could be perceived, were the error multiplied a million 
 fold? 
 
 JB. — None whatever, that I know of. 
 
 A. — What, then, would be the harm of introducing a supposition 
 which would save much trouble, and do no mischief? 
 
 B. — I am not aware that I admitted such a supposition would do 
 no mischief, when I said that it would not sensibly vitiate the appli- 
 cation of mathematics to what are commonly called the arts of life. 
 I see that your idea of mathematics is very much like that which a 
 shoemaker has of his tools. If they make shoes v<rhich keep the 
 weather out, and bring customers, he need not wish them to do more, 
 or inquire further into any use, actual or possible, which they may or 
 might have. The end he proposes to himself is answered, when he 
 has sewed the upper leather firmly to the sole. But whether his art 
 serves any higher purpose — whether the possibility of obtaining con- 
 veniencies, and avoiding hardships (which it creates in one respect), 
 excites industry and ingenuity, creates property to equalise the 
 fluctuations of harvests and commerce, and prevent the community 
 from undergoing periodical pests and famines — makes men so de- 
 pendent on each other that internal war is next to impossible, and 
 external war a grave and serious consideration, &c. &c., are not 
 matters for the thoughts of a working shoemaker ; nor will similar 
 considerations ever enter the mind of a working mathematician. You 
 have spoken of the purposes of life ; I do not know what the purposes 
 of your life may be, but if among them you count such a discipline 
 of the mind as may always render your perception of the force of an 
 argument properly dependent upon the probability of the premises, 
 and the method by which the inferences are drawn, it will be one of 
 your first wishes to propose to yourself, as a standard and a model, 
 some branch of study in which the first are self-evident, or as evident 
 as any thing can be, and the second indisputable and undisputed. 
 For though you may find no other science which will compete with 
 this in accuracy, yet you will be more likely to infer correctly, when 
 
NUMBER AND MAGNITUDE. 45 
 
 you have seen what you know to be correct inference, than you would 
 have been if you had never, in any case, distinguished between de- 
 monstration of certainties and presumptions from probabilities. And 
 still more, will you be qualified to refute, and refuse admission to, 
 that which takes the form of accuracy without the reality. If the 
 mathematical sciences be good as a weapon, they are a hundred fold 
 better as a shield. I have seen many who were visibly little the 
 better for their mathematical studies in what they advanced ; but very 
 few indeed who were not made sensibly more cautious in what they 
 received. 
 
 A. — But is not my notion adopted in practice by a great part of 
 the mathematical world, particularly on the continent. 
 
 B. — It is certainly true, and it is particularly the case with the 
 French, who, though they have done more than any other nation, since 
 the time of Newton, to advance the mathematical sciences, have been 
 by no means anxious to consider them as resting on other evidence 
 than that — not of the senses — but of the limits of the senses. One 
 of their most celebrated elementary writers considers none but arith- 
 metical proportion, and begins his work by shewing either that two 
 straight lines have a common measure, as in page 12, or that the 
 remainder " echappe aux sens par sa petitesse." All his propositions, 
 therefore, in geometry., are either true, or so nearly true, that the 
 difference is imperceptible. The phrase we have quoted is an honest 
 and a valuable admission ; it shews you, that in the opinion of one 
 of the most useful and extensive elementary writers that ever lived, 
 arithmetical proportion makes geometry a science of approximate, 
 not absolute, truth. 
 
 A. — I see as much; but cannot the slight shifting of one of the 
 quantities which I proposed be somehow or other corrected, so as to 
 make a strict and useful theory of the proportions of incommensurable 
 quantities ? 
 
 B. — Yes, and in a very simple way; by adopting the definition 
 of Euclid. This may surprise you, but I will soon shew that the 
 most natural correction of your notion leads direct to the definition of 
 Euclid. Let it be granted that A and B being commensurable, and 
 mA=nB, proportion between A, B, P, and Q means that mP=nQ. 
 Now you want, when A and B are incommensurable, to be allowed to 
 substitute B + K instead of B, where K is excessively small. I 
 suppose you would be perfectly content if it could not be made 
 
46 CONNEXION OF 
 
 visible by any microscope. Now I am of a somewhat more abstract 
 turn, and should not like my geometry to be put in peril by the 
 abolition of the excise on glass ; which it might be by the allowance 
 of experiments for the improvement of that article, which are now 
 effectually prevented. I cannot admit B-fK, where the magnitude 
 I w^ant to reason upon is B. But as the definition of proportion of 
 incommensurables is not yet settled, let us examine this case : A and 
 B being incommensurable, let P and Q be quantities of such a kind 
 that A and B -f K are commensurable, and also P and Q + Z, and 
 that the four just named are arithmetically proportionals. Let it be 
 possible, these conditions subsisting, to make K and Z as small as 
 we please : not as small as this, that, or the other small quantity, but 
 smaller than any whatsoever which may be named, being still some 
 quantities. You wish to substitute B + K and Q + Z for B and Q : 
 I prefer to use the conditions laid down to ascertain how B and Q 
 themselves stand related to A and P. Let us suppose we name two 
 small magnitudes, K' and Z', of the same kind as A and P, or B and 
 Q, or K and Z, which we are at liberty to make as small as we 
 please. We can then find K and Z less than K' and Z', and such 
 that A, B + K, P, Q 4- Z are proportional. Suppose A and B + K 
 commensurable, and let 
 
 mA. = w(B + K) whence m? = w(Q + Z) 
 
 whence it is easily proved, as in page 24, that the relative scale of 
 multiples of A and B + K is the same as that of P and Q + Z. I 
 say it follows, that the relative scale of A and B is the same as that 
 of P and Q ; for, if not, the two latter scales must differ somewhere. 
 Let it be that vA is greater than tuB, but v P less than toQ. Then, 
 since vA is greater than t(;B, let K be taken so small (it may be as 
 small as we please) that vA shall also exceed w;(B +K), whence, by 
 the proportion assumed in the hypothesis, vP exceeds w(Q + Z), 
 while, by the hypothesis we are trying, vP is less than wQ. This is 
 a contradiction, for uP cannot exceed wQ-f-tuZ and fall short oiwQ, 
 at the same time. In the same way, any other case may be treated ; 
 and it follows that our suppositions, if K may be as small as we 
 please, amount to an hypothesis from which Euclid's definition fol- 
 lows. If, in the above, we suppose vA less than w;B, while uP is 
 greater than lyQ, we see that vA also falls short of v(B -fK), whence, 
 
NUMBER AND MAGNITUDE. 47 
 
 by the proportion, vP falls short of v(Q + Z), which cannot be if it 
 exceed v Q. 
 
 A. — But is not this deduction, namely, Euclid's definition, more 
 cumbrous than the form from which it has just been deduced? 
 
 JB.-— Howso? 
 
 A. — Does it not involve an infinite number of considerations, ex- 
 tending the whole length of the multiple scales ? 
 
 B. — And does not your definition do the same thing, unless you 
 stop somewhere with the values of K and Z ? Is it not necessary, if 
 we would not be merely microscopically correct, but absolutely cor- 
 rect, to suppose that K and Z may be diminished and diminished ad 
 infinitum ? And what difference is there, as to the number of con- 
 siderations in question, between two magnitudes which are to diminish 
 without limit, and a set of increasing multiples of two given mag- 
 nitudes ? 
 
 A. — But Euclid's definition seems to wander such a way from the 
 quantities in question, while the other remains close to them, and we 
 never seem to quit them, except for something very near to them. 
 The actual application of the definition I prefer will require nothing 
 but the division of all magnitudes into aliquot parts. 
 
 J5. — Your objection amounts to this ; that you feel the fractions 
 of a quantity to be more closely connected in your mind with the 
 quantity itself than its multiples. This may be the case ; and, if so, it 
 is some reason for preferring the form to which you seem most inclined. 
 But there may be a stronger reason for preferring the other ; and, 
 undoubtedly, as long as difficulties exist, every system of science must 
 be a balance of inconveniences. But Euclid is, of all men who ever 
 wrote, the one who has a reason for the course which he takes, where 
 there are two or more. I suppose you cannot but admit that it is 
 better to found a definition in geometry upon the result of something 
 which can actually be done, bi/ the means of geometry , than upon 
 something wliich can only be conceived or imagined to be done, with 
 what certainty soever ; for instance, you would not wish to be obliged 
 to use other means than the straight line and circle, or to suppose an 
 object gained without using any means at all ? 
 
 A. — Certainly not. That there shall be no assumption of me- 
 chanical power beyond that of drawing a straight line or circle, is the 
 foundation of pure geometry. 
 
 B. — Tlien the question is settled in favour of Euclid's definition j 
 
48 CONNEXION OF 
 
 for, without either assuming more mechanical means, or making a 
 gratuitous assumption, no angle, nor arc, nor sector of a circle, can be 
 divided into 3, or 6, or 9, &c. parts, unless it be a right angle, or a 
 given half, fourth, eighth, &c. of a right angle. There are some other 
 exceptions ; but, generally, to cut any angle into three equal parts is 
 a geometrical impossibility, and certain algebraical considerations 
 fuiTiish the highest presumption that it will always remain so. 
 
 A. — But this difficulty is still left: how are we ever to shew 
 that there are such things as proportional quantities ? 
 
 B. — We can do this so easily, that the greatest stumbling-block 
 of the process lies in its being so easy and perceptible, that a beginner 
 does not very well see where lies the knowledge he has gained, unless 
 he has paid profitable attention to the definition of proportion. From 
 the first book of Euclid it is evident that a rectangle is doubled by 
 doubling tl»e base, trebled by trebling it, and so on ; and also, that of 
 two rectangles between the same parallels, the greater base belongs to 
 the greater, and the lesser base to the lesser. Now, let B and B' 
 represent two bases, and R and R' the rectangles upon them, the 
 altitudes, or distances of the parallels, being the same. If then we 
 take the first base m times, giving m B, the rectangle upon that base 
 is m R : if we take the second base n times, giving n B', the rectangle 
 upon that base is wR', the parallels always remaining the same. 
 Hence it follows, that twB and wB' are bases to the rectangles wR 
 and nR' between the same parallels; accordingly, therefore, as w/B 
 is greater than, equal to, or less than n B', so is mR greater than, 
 equal to, or less than n R' : and this being true for all values of m 
 and n, it follows that B has to B' the ratio of R to R', or the bases 
 of rectangles between the same parallels, and the rectangles them- 
 selves are proportionals. 
 
 A. — Am I to understand then that there are difficulties in the 
 way of considering magnitude in general, which are not found in 
 arithmetic, or the science of abstract number ? 
 
 B. — Quite the reverse : the difficulty arises from the deficiences 
 of arithmetic itself, and from their being ratios which the ratio of 
 number to number cannot represent. 
 
 A. — But how is that? arithmetic always seemed clear of such 
 difficulties as we have been considering. 
 
 B. — And so would this subject, if the disposition to be satisfied 
 with what is in the book, which is part and parcel of almost every 
 
NUMBER AND MAGNITUDE. 4& 
 
 beginner had been permitted to rest quietly upon a theory of com- 
 mensurable ratios. But did you never, in arithmetic, hear of the 
 creation of a nonexistent number or fraction, in spite of there being 
 no such thing, by agreeing that there should be such a thing, and 
 drawing a picture to represent it? 
 
 A. — I do not understand the jest; but I suppose you allude to 
 algebra, and to quantities less than nothing ? 
 
 B. — Not at all ; lam speaking of pure arithmetic- To me, — 2 
 is a much easier symbol, or picture, than 's/2; and even the difficulties 
 of \/ — 2 lie as much in the \/ as in the — . 
 
 A. — But I do not understand what you mean by saying that \/2 
 does not exist; it is the square root of 2, and multiplied by itself it 
 gives 2. You may find it as nearly as you please. 
 
 B. — If it be the object of arithmetic, commonly so called, it is 
 either a whole number or a fraction. Which of these is it ? 
 
 A. — It is a fraction; 1-4142136, very nearly. 
 
 JB. — I did not ask you what it is veri/ nearly/, but what it is ? 
 
 A. — It cannot be given exactly, but we all know there is suclva 
 thing as the square root of 2. 
 
 JB. — If the objects of arithmetic were numbers, fractions, and 
 things, and the latter term had a definition, I might admit what you 
 say. And in concrete arithmetic, where 1 is a thing, a foot, a 
 pound, or an acre, I admit that there is such a thing as s/2. But 
 that thing is not attainable arithmetically by taking any aliquot part 
 of the thing 1, and repeating it any number of times. In abstract 
 arithmetic the square root of 2 is an impossibility ; and having no 
 existence, I do not see how one fraction can be said to be nearer to 
 it than another, except in this sense, that 2 + z may be made to have 
 a square root where z may be less than any fraction we name. The 
 independent existence of v 2 is an algebraical consideration of some 
 difficulty; that is, belongs to the science which has relations of 
 symbols, under prescribed definitions, for its object, without reference 
 to their numerical interpretation. The difficulties of \/2 are precisely 
 those of incommensurable magnitudes; in fact \/2 is the diagonal 
 of a square whose side is 1. But it is to algebra that difficulties of 
 this kind should be referred. The student, if he use s/o, in pure 
 arithmetic, must expressly understand it as a fraction whose square is 
 nearly 2, and must consider this part of arithmetic (without algebra 
 
 F 
 
50 CONNEXION OF 
 
 as a science of approximation, unless geometry, or some other science 
 of concrete quantity, be supposed to lend its aid. 
 
 A. — But I cannot divest myself of the idea that n/2, n/s? and 
 V 6 are really fractions, and that the product of the two first gives 
 the last. I suppose, in some sense or other, you admit this pro- 
 position ? 
 
 B. — Certainly. If n/2 + j' and n/3-|-j/ and n/6+2 be made 
 to exist, by giving proper values to x, y, and z, which may all be as 
 small as I please, and if, moreover, x, 3/, and z be so related that 
 2 = 3 T + 2 j/ + xi/j which condition does not interfere with the last ; 
 I can then admit that 
 
 But I do not allow myself to suppose that (understanding by multi- 
 plication the taking of one number or fraction as many times or parts 
 of times as there are units or fractions of a unit in another), there can 
 be such a truth as that 
 
 ^2 (neither number nor fraction) 7WMZ^/p/ifd by VS (do. do.) = n/6 (do. do 
 
 But this is beyond our subject, except so far as it shews that the 
 difficulty lies more in arithmetical than in geometrical considerations. 
 
 A. — Might we not then dispense with arithmetic altogether, and 
 make a definition corresponding to proportion for geometry? 
 
 B. — Yes; but the difficulty would appear in another shape, of 
 the very same substance. Let four lines be called proportional when, 
 being straightened without alteration of length, if necessary, the 
 rectangle made by the first and fourth is equal to that made by the 
 second and third. Let areas be proportional when, being converted 
 into rectangles with a common altitude, their bases are proportional. 
 Let angles be proportional, when they are angles at the centre of 
 proportional arcs of the same circle. But here would immediately 
 arise this difficulty, — to make a straight line equal to a given arc of 
 a circle ; which is out of the power of the geometry of straight lines 
 and circles. 
 
 A. — Is not the reductio ad ahurdum (which is very much used 
 in the establishment of the theory of proportion) rather a suspected 
 method. I have heard it called indirect demonstration ; and it is 
 frequently stated as a defective method, not to be used if it can 
 possibly be avoided. 
 
NUMBER AND MAGNITUDE. 51 
 
 jB. — The complaints against this method of demonstration have 
 become much more frequent, if not entirely made their appearance, 
 since the time when logic was a necessary part of a liberal education, 
 as it once was, and as I hope it will be again. I have sometimes 
 wondered whether this argument would have been considered objec- 
 tionable if it had been reduced to the form " A is B, B is C, therefore 
 A is C ;" as follows : " Every contradiction of P is a contradiction 
 of the proposition that the whole is greater than its part ; but every 
 contradiction of this proposition is false : therefore every contradiction 
 of P is false ; or P is true." The reductio ad ahsurdum is as conclu- 
 sive, and may be made as intelligible, as any x)ther argument. And 
 if any argument be good in proportion to the effect upon the mind, 
 where is the affirmative proposition, in geometry or not, which the 
 mind seizes as readily as it recoils from an absolute contradiction in 
 terms ? Where is the likeness or resemblance between things which 
 are alike, that is so forcible as the unlikeness or want of resemblance 
 of two ideas which palpably contradict, such as black is white ? 
 
 A. — Is there then no advantage in the direct over the indirect 
 demonstrations ? 
 
 B. — D'Alembert has said that the former are to be preferred 
 " parce qu'elles ^clairent en mIme-temps qu'elles convainquent," which 
 is a good description of the difference. But even this must be taken 
 with some allowance, for there are many indirect demonstrations 
 which are highly instructive. 
 
 'Recapitulation. By the ratio of A to B, we mean (without any 
 further specification at present) a relation between the magnitudes of 
 A and B, determined by the manner in which the multiples of A are 
 distributed, if each be written between the nearest multiples of B in 
 magnitude. That is, if B, 2 B, 3B, &c., be formed, and A, 2 A, 
 3 A, &c., and if A lie between B and 2B, 2 A between SB and 4B, 
 and so on, the relative scale 
 
 B, A, 2B, 3B, 2 A, 4B, &c. 
 
 is to be the sole determining element of the ratio, so that there is to 
 be nothing but the order of this scale on which the ratio depends. 
 And if P and Q be two other magnitudes with the same order in 
 their scale, P compared with A, and Q with B, then A and B are to 
 be said to have the same ratio as P and Q. But if any multiile of A 
 
52 CONNEXION OP 
 
 precede among the multiples of B the place which the corresponding 
 multiple of P occupies among the multiples of Q, then A is to be 
 said to have to B a less ratio than P has to Q. But if a multiple of 
 A come later in the series of multiples of B than its corresponding 
 multiple of P in the series of multiples of Q, then A is said to have to 
 B a greater ratio than P has to Q. It is plain that the ratio of A to 
 B must be greater than, equal to, or less than that of P to Q, and 
 also, that in saying A is to B as P to Q, we also say that B is to A 
 as Q to P. 
 
 [We must remind the student that we have now nothing to do 
 with the reasons of this definition, or the accordance of its parts with 
 each other, or with any notion of ratio more than is contained in it. 
 We are merely now concerned to know what follows from this 
 defintion. The numbering of the following propositions is that in 
 Euclid.] 
 
 When A has to B the same ratio as P to Q, the four are said to 
 be proportionals, and are written thus : 
 
 A : B:: P : Q 
 
 which is read A is to B as P is to Q. 
 
 IV. If A : B : : P : Q | m and w being any 
 
 Then mA : wiB::wP : nQ) whole numbers. 
 
 This we know when we see that any quantities being arranged in 
 order of magnitude, so will be their multiples. If the scales bQ 
 
 B A 2B 3B .....^ 
 Q P 2Q 3Q 
 
 the following scales 
 
 mB mA 2mB 3mB ...... 
 
 kQ wP 2nQ, 3wQ 
 
 will also be arranged according to magnitude. Whence the pro- 
 position. 
 
 VIT. IfA, B, C, be three homogeneous magnitudes (all lines, or 
 all weights, &c.)and if A = B ; then 
 
 A : C :: B : C 
 and C : A :: C : Bj 
 
 for the scales must evidently be identical, 
 
NUMBER AND MAGNITUDE. 53 
 
 VIII. A-fM has a greater ratio to B than A has to B, and B has 
 a less ratio to A+M than B has to A. Let M be multiplied so many 
 times that it exceeds B ; say wM = B + K : then 
 
 7W(A + M) = TwA + B + K 
 
 Let w A lie between vB and (u + l) B ; then 7w(A + M) lies between 
 uB + B + Kand (v + 1)B + B + K, and certainly beyond (u-|-l)B. 
 Consequently, in the scales of A-fM and B, and A and B, a multiple 
 of A + M is found to be in a higher place among the multiples of B 
 than the same multiple of A among the multiples of B. Whence, by 
 definition, A4- M has to B a greater ratio than A to B. The second 
 part of the proposition is but another way of stating the first, as 
 appears from definition. Thus we may also say that A has to B a 
 less ratio than A + M has to B. 
 
 IX. If A : C::B : C then A = B 
 or if C : A::C : B then A = B 
 
 For (viii.), if A be greater than B, A has to C a greater ratio than 
 B to C, which is not true. If A be less than C, A has to C a less 
 ratio than B to C, which is not true : therefore A=B. The same 
 reasoning proves the second case. 
 
 X. If A have to C a greater ratio than B has to C, then A is 
 greater than B. For if A were equal to B, then these ratios would be 
 the same: if A were less than B (viir.), then would A have to C a 
 less ratio than B has to C. Therefore, A is greater than B. Similarly, 
 if A have to C a less ratio than B has to C, A is less than B. And 
 if C have a greater ratio to A than to B, A is less than B ; if C have 
 a less ratio to A than to B, A is greater than B. 
 
 XI. If the ratios of C to D and of E to F, be severally the same 
 as that of A to B, then C has to D the same ratio as E to F. This 
 answers to a case of the general axiom, that two things which are 
 perfectly like to a third in any respect, are perfectly like each other 
 in that respect. The multiples of C are distributed among those of 
 D in the same manner as those of A among those of B, as are ihose 
 of E among those of F. Therefore, the multiples of C are distributed 
 among those of D as are those of E among those of F. Whence tlie 
 proposition. 
 
 XII. If A be to B as C to D, and as E to F, then as A is to B 
 so is A+C + E to B+C+F. 
 
 I 2 
 
54 CONNEXION OP 
 
 For mK lying between nB and (n-f-l)B, then mC lies between 
 nD and (n4-l)D» and mE between wF and (n + l)F, and, con- 
 sequently, /n A -f- w C + w E, or w (A+C -f E) between n (B+D+F) 
 and (n + 1) (B-f D+F). Whence the proposition. 
 
 XIII. If A have to B the same ratio as C has to D, but C to D 
 a greater ratio than E to F, then A has to B a greater ratio than 
 E to F. 
 
 This is one of a class of propositions which come under this 
 general theorem : for any ratio, an equal ratio may be substituted, 
 and all consequences of the first ratio are consequences of the second. 
 This, which seems very evident, may appear so upon mistaken evi- 
 dence. Ratios, as far as we have yet gone, are not quantities, but 
 expressions of that relation between quantities upon which the order 
 of magnitude of their multiples depends. For quantity, we may 
 substitute other quantity equal to the first in magnitude wherever the 
 relation is one which depends only on quantity; we may not sub- 
 stitute a triangle of the same area instead of a square, except there be 
 question of nothing but superficial magnitude, or area. Ratio, again, 
 is to us at present the order of the multiples, so that if A and B have 
 their multiples arranged among each other in a given order, if P and 
 Q have the same, we may say that whatever is true of the order of 
 multiples of A and B, is also true of the order of P and Q ; whatever 
 connexion the order of multiples of A and B establishes between A 
 and B and other magnitudes, the same connexion exists between P 
 and Q and those other magnitudes, because the accident of A and B, 
 which is the sole connexion between them and the consequence inferred, 
 is also an accident of P and Q. The necessity for going over such 
 considerations, arises from its never being allowed to be taken for 
 granted that a mathematician has studied logic. Hence Euclid* is 
 frequently obliged to reiterate the same assertions in different forms. 
 To take the proof of the present proposition ; to say that C has to D a 
 greater ratio than E to F, is to say that mC can be found greater than 
 
 • Euclid was a contemporary of Aristotle, as is generally supposed, 
 and may, therefore, never have seen the science of the latter. It is free 
 to us to suppose that if he had, he would have distinguished between a 
 purely logical and a geometrical consequence : that is, would not have 
 reiterated the same proposition iu different forms ; or, if you please, 
 different cases of the same verbal truth as if they were distinct truths : 
 and we will suppose so accordingly. 
 
NUMBER AND MAGNITUDE. 55 
 
 nD, while w E is equal to or less than nF. But to say that A has to 
 B the ratio of C to D, is to say that whenever mCis greater than nD, 
 mA is greater than wB. Therefore, to say that the ratios of A and B 
 and C and D are the same, but the latter greater than that of E to F, 
 is to say that wA may be greater than nB, while /nE is equal to or 
 less than wF ; or that A has to B a greater ratio than E has to F. 
 
 Now, let the student compare this with the following proposition. 
 A and B are greens of exactly the same shade : but B is a darker 
 green than C, therefore, A is a darker green than C. Would it be 
 unnecessary to prove this ? then it is equally unnecessary to prove the 
 preceding. But we will prove this in the same manner as we prove 
 the preceding. Let there be a test of greenness, which decides between 
 two greens (there is a test of comparison of ratios in Euclid), and 
 apply the test to B and C. The result is, of course, that B is the 
 darker. But A being by hypothesis exactly the same as B, the testing 
 operation would be self contradictory if it did not exhibit, when 
 applied to A and C, the very same intermediate process by which 
 we were able to compare B and C, with the same result. If the 
 above-be unnecessary, then the demonstration of Euclid's proposition 
 is unnecessary. 
 
 The fact is, that there are in geometry two distinct sorts of de- 
 monstration, the first of which is only a portion of the second. The first 
 is the verbal treatment of the terms of an hypothesis, and the deve- 
 lopement of all assertions which are necessarily included in the terms 
 of the proposition, without drawing upon any other axioms or 
 theorems for evidence. It is the purely logical process, by which we 
 make two assertions put together shew their joint meaning, and 
 express what, without deduction, they only imply. Thus, from 
 " Every A is B," and " no B is C," we make it evident that in these 
 assertions is necessarily contained a third, that " no A is C." Thus 
 it has been shewn that we cannot allow simultaneous existence to the 
 two propositions, " A is to B as C is to D," and " C is to D more 
 than E is to F," without almost expressing, and certainly implying, 
 by the mere meaning of our terms, this third proposition, that " A 
 is to B more than E is to F." 
 
 The second process is that in which the demonstration, besides 
 the purely logical process of extracting implied meanings out of the 
 expressions of the hypothesis, appeals to propositions which are not 
 in the hypothesis, and which, for any thing the hypotheses tell us to 
 
56 CONNEXION OF 
 
 the contrary, may or may not be true. Of course — not logic, but — 
 reason requires that these propositions should have been previously 
 proved, or assumed on their own evidence expressly. Let us take the 
 following proposition, " The sum of the circles described upon the 
 two sides of a right-angled triangle is equal to the circle described 
 upon the hypothenuse." Now, take every notion implied in this 
 hypothesis, " Let there be a right-angled triangle, and let circles be 
 described on its three sides." The united faculties of man never 
 proved that the sum of the circles on the sides was equal to the 
 circle on the hypothenuse, without assuming with Euclid, to the effect 
 that only one parallel can be drawn through a point to a given right 
 line ; with Archimedes, to the effect that the chord of a curve is shorter 
 than its arc, &c. &c.; and various consequences. But are any of these 
 propositions necessary to our complete definition of a right angle, a 
 triangle, or a circle? If not, we have a broad and easily recognised 
 distinction between the first and second method of demonstration ; 
 the first, an operation of logic, or deduction from the premises of the 
 hypothesis ; the second, introducing premises from without. 
 
 There are two classes of reasoners whose ideas we recommend 
 the student closely to examine, before he finally decides: 1. Geome- 
 trical writers in general, who pay no attention to the methods which 
 they are using, but let the first book and the fifth book of Euclid 
 contain no difference by which it may be remarked that the processes 
 contained in the two are different acts of mind. Did they ever 
 think that geometry could be made the engine by which the student 
 could examine certain operations of his own faculties, or did they only 
 imagine that it was a method of making very sure that squares, 
 circles, &c. had such and such properties? 2. The class of metaphy- 
 sical writers, who express themselves to the effect that all mathema- 
 tical propositions are contained in the definitions and axioms, in a 
 sense in which other results of reasoning are not. Put them to the 
 proof of this assertion as to geometry, and then as to arithmetic. 
 
 The whole of the process in the fifth book is purely logical, that 
 is, the whole of the results are virtually contained in the definitions, 
 in the manner and sense in which metaphysicians (certain of them) 
 imagine all the results of mathematics to be contained in their 
 definitions and hypotheses. No assumption is made to determine the 
 truth of any consequence of this definition, which takes for granted 
 inore about number or magnitude than is necessary to understand the 
 
NtJMBEli AND MAGNITUDE. 57 
 
 definition itself. The latter being once understood, its results are 
 deduced by inspection — of itself only, without the necessity of looking 
 at any thing else. Hence, a great distinction between the fifth and 
 the preceding books presents itself. The first four are a series of 
 propositions, resting on different fundamental assumptions ; that is, 
 about different kinds of magnitudes. The fifth is a definition and its 
 developement; and if the analogy by which names have been given 
 in the preceding books had been attended to, the propositions of that 
 book would have been called corollaries of the definition. 
 
 XIV. If A be to B as C is to D, all four being of the same kind, 
 then if A be greater than C, B is greater than D ; if equal, equal, and 
 if less, less. 
 
 A must either be > =: or < C. Let A be greater than C ; then 
 mA is greater than mC. Let wA lie between wB and (n-|-l)B; 
 then will mC lie between nD and {n-\-\)T). But because A exceeds 
 C, 2A exceeds 2C by twice as much, &c., and mA exceeds mQ by 
 m times as much ; or rnA may be made to exceed mC by a quantity 
 greater than any one named, say greater than B and D together. 
 Then the order of magnitude of the four multiples mC (n+OD, wB, 
 wA must be as written: for (n-|-l)D does not exceed rnC by so 
 much as D, and wB does not fall short of ;n A by as much as B, 
 while mK exceeds wC by more than B and D put together. There- 
 fore, nB is greater than (n+l)D, and still more than nD. That is, 
 B is greater than D. 
 
 Let A be equal to C. If B exceed D at all, twB may be made to 
 exceed mVi by more than D, or twB may be made, from and after 
 some value oim, greater than (m -j- 1)D. That is, the order of mag' 
 nitude may be made 
 
 ml> (m + l)D . mB (m+l)B 
 
 Having gone so far on the scales that this order becomes per^ 
 manent, go on till a multiple of C (/cC) falls between the two first. 
 Then, by the definition, kK falls between the two last, which is ab- 
 surd ; for, because A = C, /cA = /cC ; therefore, B does not exceed D. 
 In the same way it may be shewn that B does not fall short of D. 
 Therefore, B=:D. 
 
 The remaining case (A less than C) may be proved like the first. 
 
 XV, A is to B as mk is to mB 
 
58 CONNEXION OF 
 
 The scale of multiples of A and B is nowhere altered in the order ot 
 magnitude by multiplying every terra by w. If p A lie between qB 
 and (^-t-l)B, (pin) A which is p{m A) lies between ^(mB) and 
 (y + l)(mB). 
 
 XVI. If A be to B as C is to D 
 and if all four be of the same kind, 
 
 Then A is to C as B is to D. 
 
 (iv.) mA isto mB as nC isto wD 
 
 (xiv.) If wA be greater than nC, wB is greater than nD, if 
 equal, equal ; if less, less. Therefore, A is to C as B to D. 
 
 XVII. If A + B be to B as C + D to D, then A is to B as C is 
 to D. If ;« A lie between n B and (n + 1) B, it follows that mA+m B, 
 orm(A-f B) lies between (m + n)B and (m-\-n-\-l)B. Then, by 
 the proportion, m(C + D) lies between {m + n)D and (w + n + 1)D, 
 ormC + wD lies between mD + wD and wD + (n + l)D, or wC 
 lies between nD and (n + l)D. Therefore, the scales of A and B, 
 and of C and D, are the same ; whence the proposition. 
 
 XVIII. If A be to B as C is to D, then A -f B is to B as C + D 
 is to D. A proof of exactly the same kind as the last should be given 
 by the student. 
 
 XIX. If A : B : : C : D, C and D being less than A and B, then 
 A : B : : A — C : B — D. For the hypothesis gives A to C as B to 
 D, and A is C + (A — C), and B is D + (B — D), whence, 
 
 C + (A — C) isto C as D + (B — D) isto D 
 (xvir.) A — C isto C as B — D isto D 
 
 (xvi.) A — C isto B — D as C to D, or as A to B 
 
 XX. If A be to B as D to E 
 and B to C as E to F 
 
 S greater than f fgreater than 1 
 
 equal to > C, when B is -J equal to > V 
 less than J U^ss than J 
 
 Let A be greater than C ; then A is to B more than C is to B ; 
 but A is to B as D to E, and C to B as F to E ; therefore, D is to E 
 more than F is to E, or D is greater than F. In a similar way the 
 other cases may be proved. 
 
NUMBER AND MAGNITUDE. 69 
 
 Hence it follows, that A is to C as D to F. For, 
 
 (vi.) 7»A is to wB as mD to twE 
 
 wB is to wC as nE to nF 
 therefore, wA is >= or <nC when mD is >= or <nF 
 whence, A is to C as D to F. 
 
 XXI. If of the magnitudes 
 
 ABC ^ A:B::E:F 
 
 we have 
 D E F B:C::D:E 
 
 Then A > = or < C when D > = or < F 
 
 Let A be greater than C ; then A is to B more than C is to B : as 
 before E is to F more than E is to D, or D is greater than F. Simi- 
 larly for the other cases. 
 
 XXII. If there be any number of magnitudes, 
 
 A B C D 
 
 P Q R S 
 
 and if any two adjoining be proportional to the two under or above 
 them, then any two whatsoever are proportional to the two under or 
 above them. For, since (xx.) 
 
 A:B::P :Q' 
 B :C::Q : RJ 
 
 Hnt- I ', ? n ! : K. ! S » 
 
 &C. 
 
 J|Therefore,A:C::P:Rl_. . a r» o c 
 IJ ^ ' _ _ „ > Therefore, A : D : : P : S, 
 
 ^ But, C :D::R: Sj 
 
 XXIII. In the hypothesis of (xxi.), by proof as before in (xx.), 
 A is to C as D to F. 
 
 XXIV. If A be to B as C to D '^^ then A + E is to B 
 
 D 
 
 and 
 
 E be to B as F to D J as C + F lo 
 
 For, ^ 
 and 
 
 b;1;;d:f} ">--' ^■■^-■''■■' 
 
 
 (xviji.) A-}-E:E::C4-F:F 
 
 
 But, E : B : : F : D 
 
 
 Therefore, A + E:B;:C-fF:D 
 
 XXV. If A : B : : C : D, all being of the same kind, the sum 
 of the greatest and least is greater than that of the other two. First, 
 which are the greatest and least ? If A be the great^t, then C is 
 greater than D ; and because A : C ; *. B : D, B is greater than D ; 
 
60 CONNEXION OF 
 
 therefore, D is the least. Now, prove that if B be the greatest, C is 
 the least; and that, by inverting the proportion, if necessary, it may 
 always be written with the greatest term first, and the least last. 
 
 When A is the greatest, since A — B : A : : C — D : D, A — B 
 is greater than C — D ; therefore, ( A — B) + B + D is greater than 
 (C — D) 4- B + D, or A 4- D is greater than C + B. 
 
 If there be a given ratio, that of A to B, and another magnitude 
 P, there must be a fourth magnitude Q, of the same kind as P, such 
 that A is to B as P to Q, or Q to P as B to A. 
 
 Firstly; Q may certainly be taken so small that (mB being 
 greater than nk) mQ shall be less than wP. Find m and n to 
 satisfy the first conditions, and let K satisfy the second. Then K is 
 to P less than B is to A. Now (mB being less than 7iA), Q may be 
 taken so that mQ shall be greater than nP. Find m and n to satisfy 
 the first, and let L satisfy the second. Then, L is to P in a greater 
 ratio than A to B. And it is immediately shewn that every magni- 
 tude less than K is to P less than B to A, and every magnitude 
 greater than L is to P more than B to A. Whence, it is between 
 K and L that the fourth proportional Q is found, if any where. 
 There cannot be more than one such value of Q ; for, if there be two 
 different magnitudes V and W, since, then, by taking m sufficiently 
 great, we may make mV and m W differ by more than P, it is impos- 
 sible that both wV and mW can lie between the same consecutive 
 multiples of P, as those of B which contain between them mA. And 
 the above also evidently shews, that if we suppose a magnitude Q, 
 changing its value from K to L, it cannot during its increase become 
 of the same kind as L, namely, more to P than B is to A, and then 
 again become of the same kind as K. For, whatever magnitude has 
 this property of L, every greater one has the same. There is then 
 only one point between K and L at which this change takes place, 
 and we have, therefore, this alternative: Either G (between K 
 and L) is less to P than B is to A, and every magnitude greater than 
 G is more ; or, some magnitude G between K and L is the same to 
 P as B to A, and is the intermediate limit lying above all those 
 which are less to P, and below all those which are more. By dis- 
 proving the first alternative, we prove our proposition. If possible, 
 let G be less to P than A to B, G+V more, however small V may be. 
 
NUMBER AND MAGNITUDE. 61 
 
 Then maymG be made less than nP (;w A being greater than nB), 
 while m{G-\-V) is greater than wP. For the ascending assertion 
 must be converted at least into a stationary one. Let m G fall 
 short of wP by Z ; then V may be taken so small that mV shall 
 not be so great as Z, or mG+mV not so great as wG+Z, that is, 
 not so great as mP. But the first clause of the alternative supposes 
 that w(G+V) must be greater than wP, how small soever V may be; 
 therefore this clause cannot be true, or the second must be true. 
 
 This fourth proportional to A, B, and P, then, must exist; but 
 whether it can be expressed by the notation, or determined by the means 
 of any science, is another question. It can be expressed in arithmetic 
 when A and B are commensurable : it can be found in geometry 
 (by the straight line and circle) when A and B are lines or rectilinear 
 areas. But if they be angles, arcs of circles, solids, &c. it cannot be 
 assigned by the straight line and circle, except in particular cases. 
 
 Let us suppose the ratio of A to B given, that is, not A and B 
 themselves, but only the answer to this question for all values of w, 
 ** Between what consecutive multiples of B lies mA. V Suppose also 
 the ratio of B to C given ; how are we to find the ratio of A to C, or 
 can it be found at all ? that is, is it given or determined by the two 
 preceding ratios. Take any magnitude P, and determine Q so that 
 P is to Q as A to B, and then determine R so that Q is to R as 
 B to C. Then the ratio of P to R (page 59) is that of A to C ; not 
 that P is A or R is C (for they may even be magnitudes of different 
 kinds), but P is to R as A is to C. 
 
 The process by which the ratio of A to C is found by means of 
 those of A to B and B to C, is called by Euclid composition of these 
 ratios; or the ratio of A to C is compounded of the ratios of A to B 
 and B to C. What, then, ought to be meant by the ratio compounded 
 of the ratios of A to B and X to Y. Our guide in the assimilation 
 of processes, and the extension of names, is always the following 
 axiom. 
 
 Let names be so given, that the substitution of one magnitude 
 for another equal magnitude shall not change the name of the process ; 
 and, generally, that the same operations (in name) performed upon 
 equal magnitudes, shall produce the same result. 
 
 Let X be to Y as B to N, where N is a fourth proportional to be 
 determined. Then the ratio of A to N is that compounded of A to B 
 and B to N, and is what must be meant by that compounded of 
 
62 CONNEXION OP 
 
 A to B, and X to Y. It is proved in Prop. 20, that ratios com- 
 pounded of equal ratios are equal ratios. 
 
 Again, to find the ratio compounded of the ratios of A to B, 
 C to D, and E to F ; let the process by which the ratio of A to D is 
 derived from those of A to B, B to C, and C to D, still be called 
 composition. Then take B to M as C to D, and M to N as E to F : 
 the ratio of A to N is that compounded of the three ratios. 
 
 In the beginning of this work, we deduced the necessity for con- 
 sidering incommensurables in some such manner as that of Euclid, 
 from the notion which, as applied to commensurables, admits of a 
 definite representation, derived from the idea of proportion. But the 
 method of the fifth book is different. It is there implied, that where- 
 ever two magnitudes exist, their joint existence gives rise to a third 
 magnitude, called their ratio, of which magnitude no conception is 
 given except what is contained in certain directions how to apply the 
 terms equal, greater, and less, to two of the kind. On this the 
 natural question is, what sort of magnitude is this, and how do we 
 know that there is any magnitude whatsoever which admits of this 
 apparently arbitrary exposition of definitions ? This question is very 
 much to the point, and the want of an answer at the outset is a main 
 cause of the difficulty of the Fifth Book. The answer implied in the 
 work of Euclid is this : Let us first consider what will follow if there 
 be such things as ratios, or magnitudes to which these definitions of 
 equal, greater, and less apply ; we shall then shew (in the Sixth Book) 
 that there are different pairs of magnitudes, of which it may be said 
 that they have ratios,- and we shall never have occasion to inquire 
 what ratio is. 
 
 We may take a case parallel to the preceding from the First Book. 
 The notion of a straight line suggests nothing but length ; that of two 
 straight lines which meet, suggests a relation, which we may conceive 
 stated in this way. If A, B, C, and D, be straight lines, of which 
 A and B, and C and D, meet ; let A and B be said to make the 
 same angle as C and D, when, if A be applied to C, and B and D 
 fall on the same side, B and D also coincide : but let A be said to 
 make a greater angle with B than C with D, when, in a similar case, 
 B falls outside of C and D, &c. To this it would be answered, that 
 the preceding definitions are a circuitous way of saying that the angle 
 made by two lines is their opening or inclination ; an indefinite term, 
 which, though it distinguishes angle from length; does not serve to 
 
NUMBER AND MAGNITUDE. 63 
 
 compare one angle with another. And just in the same manner, if it 
 were not that the definition is more complicated, and refers to an 
 abstract, not a visible or tangible, conception, it would immediately 
 be seen that ratio is relative magnitude, — a term which is sufficient 
 to distinguish the thing in question from absolute magnitude, but 
 which does not give any means of comparing one thing of the kind 
 with another, Tiie immediate deduction of this idea is as follows : 
 If, whenever mA lies between n B and (n +1) B, it also happens that 
 mP lies between nQ and (n -f-1) Q, it follows that A, lying between 
 
 two certain fractions of B, — B, and — — B, then P lies between 
 
 n 
 the sam« two fractions of Q. Or, if wA = wB, that is, if A= — B, 
 
 then P is the same fraction of Q. Or we may state it thus : if B be 
 made unity, for the measurement of A, and Q for the measurement of 
 P, then A and P are the same numbers or fractions of their respective 
 units. 
 
 Euclid has commenced the subject with a rough definition, as we 
 have seen, p. 29, and the translators have spoiled it, by not distin- 
 guishing between quantity, and relative quantity; that is, by so 
 wording the definition as to say nothing more than that ratio is a 
 relation of magnitudes with respect to magnitude. 
 
 We now come to consider the application of the preceding notions 
 to arithmetic. Let us first separate all that part* of arithmetic which 
 relates to abstract and definite numbers, from the rest, and let us call 
 it primary arithmetic. A little observation will shew that abstract 
 number as distinguished from concrete, is really the same thing as 
 ratio of magnitude to magnitude. What is threCf for example? It is 
 an idea which we obtain equally from looking at 
 
 and 
 
 From putting such concretes together, we bring away a notion of there 
 being the same relative magnitudes existing between the individuals 
 
 • The whole of the First Book of mj Treatise on Arithmetic, with th« 
 exception of $ 158, 165-169. 
 
64 CONNEXION OF 
 
 of each pair. In the first, it is repetition^ in the second, it is length, 
 in the third, it is opening, we are reminded of; but in all three, we say 
 the first is three times the second. Now this word times is, in fact, a 
 limitation, which will not do for our present purpose; it implies that 
 we will have no other ratios except those of line to line in the series 
 
 A h 1 
 
 B h 1 1 
 
 C 1 , n ^ 
 
 D h ^ n -^ 1 &c. 
 
 made hy repetitions only : but there may be ratios which are not those 
 of line to line in any repetition, how far soever carried. 
 
 Here is a point at which we are compelled to pause, to adjust the 
 well-known terms of number to the new idea we have put upon them. 
 Abstract numbers are certain ratios; abstract fractions are certain 
 other ratios : but all possible ratios are not found among numbers and 
 fractions ; whence it arises j that primary arithmetic^ though it may be, 
 so far as it goes, a theory of ratios, is not a theory of all ratios^ nor 
 are its operations such as can be performed upon all ratios. 
 
 That ratios are magnitudes, we must have supposed firom the 
 beginning, seeing that they bear the terras equal, greater, and less. 
 But there was still this defect, that our test of A being to B more than 
 C to D, was one which left us with no idea how much more A was to 
 B than C to D; which amounts but to this, that we could not define 
 the ratio of ratios without having first defined ratio. But, in like 
 manner as arithmetic was made the guide to that notion which is 
 properly* called the ratio of incommensurable quantities, so will the 
 ratio of two ratios in arithmetic lead us, after a little consideration, to 
 the meaning of the ratio of ratios of incommensurables. 
 
 When we say two, we refer to the repetitions of the smaller in a 
 ratio of magnitudes, thus visibly related : 
 
 When we say twice two, there is a change of idiom in our language. 
 It might be, instead of twice two is four, two twos dire four ; that is, 
 where there exists that idea of relative magnitude which we signify by 
 
 • Consistently; so as to couple with operations upon problems of 
 commensurables those operations which apply to the same problems upon 
 incommensurables. 
 
NUMBER AND MAGNITUDE. 65 
 
 two, let the idea o( relation be coupled with the idea of a larger relation^ 
 in exactly the same manner as our idea of magnitude, when we look 
 
 at , is increased when we look at ; and we shall then, 
 
 by considering the result as one of relative magnitude, be led to the 
 
 idea of the relation between and . This, 
 
 of course, does not give a better comprehension of twice two is four; 
 but what it explains is, that we are using the term ratio in a consistent 
 sense, when we say that the ratio of 2 to 1, increased in the ratio of 
 2 to 1, is the same as the ratio of 4 to 1 ; and, generally, that the 
 ratio of w to 1, increased in the ratio of n to 1, is the ratio of mn to 1. 
 And the notion of relative magnitude contained in the words, ratio of 
 m top, must be the same as that contained in the words, ratio o( mn 
 to pn ; and, conversely, the notion in the latter is that implied in the 
 former. I doubt if any thing that deserves the name of proof can be 
 given of this proposition, which seems to be worthy the name of an 
 axiom. What idea we form of magnitude as portion of magnitude 
 from A and B, the same do we form from 2 A and 2B. Nor can 
 I imagine these propositions extended to fractions in any more funda- 
 mental manner than by observing, that as — taken - times is — 
 •^ ° n q nq 
 
 times (times mean times, or parts of times, either separately or both 
 
 jn p 
 
 together,) a unit, the ratio of — to 1, altered in the ratio of- to 1, 
 n 9 
 
 is the ratio of — to 1 ; or that the ratio of m to n, altered in the ratio 
 nq 
 
 of/> to 9, is the ratio o( mp to nq. These are propositions in which 
 the line between deduction and mere establishment of the synonymous 
 character of terms is very indefinite. I recommend the student to 
 examine his own idea of what he would have meant by " the pro- 
 portion of 3 to 2 increased in the proportion of 5 to 4, is the pro- 
 portion of 15 to 8." If he be a metaphysician, I refer him to his 
 oracle, on condition only that the response shall not contradict the 
 preceding proposition. 
 
 The multiplication of m and n is, then, the alteration of the ratio 
 of m to 1 in the proportion of n to 1 ; and the ratio of magnitudes 
 tn A and wA is the same as the ratio of magnitudes mB and wB, and 
 of m to n. Hence, to alter mA: nX (which is m : n) in the ratio of 
 />B to qBf which is {p : q), is the formation of mp : nq, or mp\ to 
 «jA, or mpB to w^B. Now, this is precisely what Euclid has 
 
 G 2 
 
66 CONNEXION OF 
 
 termed the composition of these ratios ; for, let w A : /tA : : vB : pB, 
 then vB : pB compounded with j^B : ^B, is vB : qB, or v : g. But 
 
 mA:nA::m:n vB : pB :: v : p 
 
 Therefore m : n is v : p or — : 1 is - : 1 
 ^ n p 
 
 V =^— V : q is ^— ' g or pm : wg' or p/wA : nqA 
 or ^?wB : wg'B. 
 
 Hence, composition is multiplication of terms, when the ratios are 
 those of number to number. Let, then, composition of ratios stand 
 for multiplication of terms, and be considered as the corresponding 
 operation in the case of incommensurable magnitudes. 
 
 Prove from this, that if U : A and U : B be compounded, giving 
 U : C, that when A=aU and B = bU, we have C=fl6U, and that 
 if U : A and B : U be thus compounded, giving U : D, we have 
 
 D = T U, D : U : : Y : 1 ; in which operations, corresponding to mul- 
 tiplication and division. 
 
 It may be a matter of some curiosity to know whether Euclid carried 
 with him the notion of multiplication of numbers in the composition 
 of ratios. In the Fifth Book, the notion of the numerical magni- 
 tude of a ratio is entirely suppressed, except only in the single word 
 TtiXiKOTyii (see page 29.) Composition* is defined to be the taking 
 an antecedent of one ratio with the consequent of another; and it is 
 not even specified that the intermediate terms are to be the same. 
 But in the Sixth Book we find composition, or collocation of ratios, 
 to mean the multiplication of their quantuplicities (see page 29). 
 
 • Ivvhirts koyov itrr) X'^-^'is tou fiyovf^.iyou fAirk rov i^of^ivov us ivos 9r^af 
 UVTO TO i^o/jctvov- — V. Def. 15. 
 
 Ao<yos iK Xoyuv evyxiltr^-a,! kiytrat ornv ai vuv koyav ^tiXixortiTts ip' 
 lavrks 'TeoX'ka.'x'kex.ffioi.a^iiacn Totuffi rtvei. — VI. Def. 5. 
 
 The second of these definitions has usually been omitted in modern 
 editions. But it is worthy of remark, that, in the first, to compound is 
 fvvriB^itr^ai ; in the second, ffvyxntr^at ; and the second is the word after- 
 wards used by Euclid, though in the sense of the first. The reason of the 
 omission appears to have been a disposition on the part of commentators 
 to consider Euclid as a perfect book, and every thing which did not 
 accord with their notions of perfection, as the work of unskilful editors or 
 interpolators. 
 
NUMBER AND MAGNITUDE. 67 
 
 The addition and subtraction of ratios can only be primarily con- 
 ceived when the latter terms of the ratios are alike. Thus, 
 
 A 
 
 B 
 
 G D 
 
 we must imagine the idea of relative magnitude given by BC com- 
 pared with A, and by C D compared with A, to be put together, in 
 order to make up the relative magnitude of BD to A. Addition and 
 subtraction are, as to ratios, ideas not so simple as multiplication and 
 division. Shew that the preceding is the only way in which m : 1, 
 increased in the ratio of w : 1, will give tww : 1, consistently with 
 the notion of multiplication of whole numbers being successive 
 additions. 
 
 When ratios have not the same consequents, they must be reduced 
 to the same consequents. Thus, A : B and C : D are added by 
 taking A : B :: P : Z and C : D :: Q : Z, and P -fQ : Z is the sum 
 of the ratios. This answers to addition of fractions. 
 
 Let P be the mean proportional between A and B, meaning that 
 A : P as P : B. It may be proved, as in page 60, that there must be 
 such a magnitude as this mean proportional, and we may also prove 
 that we can find A : P as P : Q, and P : Q as Q : B, thus forming 
 two mean proportionals. It is readily proved, that if A=aU and 
 B=6U, then P=cU where ccz=ab. If, then, a 6 be a number or 
 fraction which has a square root, P can be found commensurable with 
 A and B ; but if at have no square root, number or fraction, then P 
 is incommensurable with A and B, but not, therefore, unassignable as 
 a magnitude, though unassignable as a numerical fraction of A or B. 
 Consequently, when we speak of >/2, it must be with reference to 
 magnitude, and we mean s/l M, an accurate representative (if we 
 choose to define it so) of the mean proportional between M and 2M. 
 Similarly, when there are two mean proportionals, we find P, if 
 A=aU and B = 6U, to be cU where ccc = aby and this is incom- 
 mensurable unless a 6 be a cube number or fraction. But we may 
 
 3 /— 
 
 define v2M to be the first of two mean proportionals between M 
 and 2 M ; and so on . 
 
 Are we, then, to use long processes and comparatively obscure 
 definitions, whenever the ratios of a problem are incommensurables ? 
 By no means ; we proceed to shew that it may always be made pos- 
 
68 CONNEXION OP 
 
 sible to let the processes of arithmetic (or rather of algebra) be used 
 as if the ratios in question were commensurable ; and that we may 
 thus deduce a result which may either be interpreted strictly at the 
 end of the process, or made to give a result as near as we please to 
 the truth in arithmetical terms. Let us suppose this Problem : Two 
 pounds are spent in buying yards of stuff, and as many yards are 
 bought as shillings are given for a yard. Let x be the number of 
 yards, then x yards at x shillings a yard, gives xx shillings; whence 
 rx = 40, which is arithmetically impossible. Now, turn from num- 
 bers of pounds to quantities of silver, and let S be the silver in a 
 shilling, X that in the price given ; let L be a yard, and Y the length 
 bought. Then it is required that 40S should be given, and that X 
 should bear the same ratioto S as Y bears to L. Now, if X be given 
 for L, what must be given for Y ? Take P of such relative magni- 
 tude to X, as Y is to L ; that is, let 
 
 L: Y::X : P = 40S 
 
 But as L : Y : : S : X Therefore S : X : : X : 40S 
 
 or X must be a mean proportional between S and 40 S. Now, if we 
 make our symbols general, and let x stand for any ratio, numerically 
 possible or not, but proceed as we should do if it were arithmetical, 
 we proceed as in the first case, and find x=n/40, which, being in- 
 terpreted as a magnitude, with reference to its ratio to S, means, when 
 the symbols are general, n/40S, the mean proportional between S 
 and 40S. If we wish for an approximate numerical result, we must 
 suppose 40 + a to be the sum, where 40 + a has a square root, and 
 then we have j: = \/40+a; and since a may be made as small as 
 we please, we can make this problem as near the given one as we 
 please. 
 
 The following table should be attentively considered. In the first 
 column, an incommensurable ratio x, of X to U, is given, or a func- 
 tion of it and other ratios, under arithmetical symbols ; in the second 
 is the ratio which the function really gives, when the symbols on the 
 first side are extended in meaning. 
 
 X ox X ', 1 tlie ratio of X to U 
 
 y--y : 1 Y .. U 
 
 z.. z -.1 Z .. V 
 
NUMBER AND MAGNITUDE. 
 
 69 
 
 1 : : 1 : a: or U : X as 
 
 1 1 
 
 XX X 
 
 compounded of j: : 1 and x : \ or X : U and X ; U. Let 
 U ; X : : X : P t'nen P : X and X : U compounded give P : U 
 or i"'* : 1 is the ratio which a third proportional to U and X 
 bears to U. 
 xyz {x\\){y\\){z'.\) ratio compd. of X : U, Y : U, and Z : V. 
 
 Let X : U :: P : Y ; the above is then compounded of 
 P : U and Z : V. Let P : U : : Q : Z. The result is then 
 Q : V or xyz : 1 is Q : V 
 P a fourth prop, to U, X, and Y 
 
 Q U,P, .. Z 
 
 xy-\-yz 0-3/ : 1 is P : U when U : X : : Y : P 
 yz'.l is Q: V .... U: Y::Z:Q 
 Take Q:V::M:UorV:Q::U:M 
 P + M : U is the ratio required. 
 
 X- : 1 compd. of X : U and V : Z 
 
 z ^ 
 
 Let X : U : : P : V and P : Z is the ratio required 
 
 Now, we have assumed the operations of finding a fourth pro- 
 portional, a mean proportional, two mean proportionals, &c. Whether 
 these can be done, or whether any or all cannot be done, is a question 
 for every particular application. In arithmetic, we will suppose the 
 data arithmetical; a fourth proportional can always be found. In 
 geometry, a fourth proportional can be found to lines or rectilinear 
 areas ; but not to angles, &c. And a mean proportional cannot 
 generally be found in arithmetic, but can be found in geometry, 
 between two straight lines, or two rectilinear areas. But two mean 
 proportionals cannot be found in geometry or in arithmetic. 
 
 It must be remembered, that while we are here speaking of 
 geometry or arithmetic, we are not speaking of every conception we 
 can form of these sciences, but of the subjects as limited by the de- 
 finitions of what it has been agreed shall be called arithmetic and 
 geometry. Elementary arithmetic means the science of numbers and 
 fractions : elementary geometry, the science of space, so far as the 
 same has properties which can be deduced by allowing o^ fixed 
 straight lines and circles. To say that an angle cannot be trisected 
 geometrically f means, that it cannot be trisected by means of straight 
 
70 CONNEXION 0^ 
 
 lines and circles as defined. But there is an abundance of curves, 
 the stipulation to draw any one of wliich would secure the means of 
 trisecting an angle. And, by simply granting that a circle should be 
 allowed to roll along a straight line, and that the curve described by 
 one of its points should be granted, we can either square the circle, or 
 find the ratio of any two arcs. And, just in the same way, if we 
 were to define a journey to be 100 miles or less, it would be perfectly 
 true that we could not make a journey from London to York, but that 
 we could from London to Brighton. 
 
 It is surely time that the verbal distinction between different parts 
 of the same sciences should be done away with. Every conception 
 which can be shewn to be not self contradictory, can be as easily 
 realised by assumption as the drawing of a circle, which is itself a 
 perfect geometrical idea, and can only be roughly represented by 
 mechanical means. Whatever can be distinctly conceived, exists for 
 all mental purposes ; whatever can be approximately found, for all 
 practical uses. 
 
 It may be worth while to make the student remark the close 
 similarity which exists between the process in page 64, and that by 
 which we enlarge our ideas in algebra, from the simple consideration 
 of numerical magnitude to that of positive and negative quantities. 
 In both, we set out with a notation insufficient to express all the 
 results of problems ; in both, this circumstance is marked by the 
 appearance of unexplained results, the examination of which, on 
 wider grounds, shews the necessity for attaching more extensive ideas 
 to symbols ; and in both, the partial view first taken is wholly included 
 in the more general one : while in both, the processes conducted 
 under the wider meanings are precisely the same in form and rules 
 as those which are restricted to the original meanings of the symbols. 
 The principal difference is, that in extending arithmetic to the general 
 science of ratios, we are not engaged in interpreting difficulties 
 arising from contradictions, but from results which are only approxi- 
 mately attainable. But in both the reason is, that we set out with 
 our symbols so constructed, that we cannot undertake a problem 
 without tacitly dictating conditions to the result. In beginning 
 algebra, we make quantities indeterminate in magnitude, with symbols 
 of operation so fixed in meaning, that they cannot be used without 
 an assumption that we know which is the greater and which is the 
 
NUMBER AND MAGNITUDE. 71 
 
 less of two unkr.own quantities. We liave, therefore, to examine the 
 different cases of problems which present different results according 
 as one datum is greater or less than another; and thus we obtain those 
 extensions of meaning which will make the problems and the symbols 
 equally general. In beginning arithmetic, we invent no symbols of 
 ratio, except those which represent the ratios of magnitudes formed by 
 the repetitions of a given magnitude. These we find to be not 
 sufficient to represent ail ratios ; though it is shewn that we can make 
 them represent any ratio which magnitudes can have, as nearly as 
 we please. The invention of new symbols of ratio must require 
 the generalisation of operations ; that is, we cannot speak ofmulti- 
 plication or division of ratios generally, while these words have a 
 definition which applies only to ratios of repetitions, or commensurable 
 ratios. 
 
 There is a difference between the impossible of primary arith- 
 metic, and that of geometry. The first is unattainable by a restricted 
 definition, the second by restricting the cases of general definitions 
 which shall be allowed to be used. In arithmetic, we attempt a 
 science of relative magnitudes, by running from the general notion of 
 relative magnitude to the more precise and easy notion of the relative 
 magnitudes of one certain set of magnitudes, A, an arbitrary, A-|- A, 
 A-j-A-f-A, &c. We are very soon taught that our symbols will 
 not express all ratios, that is, if we have a general notion of ratio to 
 think about: whence our definitions are not sufficiently extensive. 
 But in geometry, having assumed notions and definitions from which 
 we cannot help conceiving an infinite number of different lines and 
 curves, we immediately proceed to cut ourselves off from the use of 
 all except the straight line and circle; that is, the straight line between 
 or beyond two given points, and the circle which has a given centre 
 and a given radial line. Until these demands or postulates are 
 looked upon as resty^ictions, their sense is never understood. (See 
 the Appendix.) 
 
 This difference is, however, not very essential ; since it is much 
 the same whether we define in too limited a manner, or whether we 
 limit ourselves to the use of only a part of a general definition. We 
 shall in the sequel discard the restrictive postulates, and suppose 
 ourselves able to draw any line which we can shew to be made by 
 the motion of a point. 
 
 The method by which Euclid first exhibits four proportional 
 
72 CONNEXION OF 
 
 Straight lines, though elegant and ingenious, has not the advantage 
 of exhibiting the notion of ratio directly applied to two straight lines. 
 The following theorem is directly proved from the first book, and 
 maybe made the guide. If a series of parallels cut off consecutive 
 equal parts from any one line which they cut^ they do the same from 
 every other. This premised, suppose any two lines OA, OB, and 
 take a succession of lines equal to OAand OB, drawing through 
 every point a parallel to a given line. Draw any other line, O C D, 
 intersecting all the parallels : from which the preliminary proposition 
 shews, that whatever multiple O a is of OA, the same is Oc of OC ; 
 and whatever Ob is of OB, the same is Oc? of OD. And if O a be 
 greater than, equal to, or less than Ob, Oc is greater than, equal to, 
 or less than, Orf. Hence the definition of equal ratios applies pre- 
 cisely to the lines OA, OB, OC, and OD, which are, therefore, 
 proportionals. This gives the construction of Book VI. Prop. 12, or 
 one analogous to it. 
 
 The metliod of finding a mean proportional between two straight 
 lines is given in Prop. 13; but as we now wish to make the straight 
 line the foundation of general conceptions of magnitude, we shall 
 pass at once to those considerations which involve any number of 
 mean proportionals. It adds considerably to the interest of this part 
 of the subject, that we are thus brought to the notions on which the 
 first theory of logarithms was founded . 
 
 Let there be any number of lines, V, Vj, V^, Vg, in con- 
 tinued proportion ; that is, let all the ratios of V to V„ V^j to Vj, Vj to 
 Vg, &c. be the same. And let V, be greater than V ; in which case 
 Vj is greater than Vj, &c. If V^ were equal to V, then would Vj be 
 equal to Vj, &c. And, first, we have the following 
 
 Theorem. By however little Vj exceeds V, the series V, Vj, &c. 
 is a series of magnitudes increasing without limit : so that, however 
 great A may be, a point may be attained from and after which every 
 term is greater than A : but in all cases whatsoever, Vj may be taken 
 
NUMBER AND MAGNITUDE. 73 
 
 SO near to V, that the terms of the series V, Vj, &c. between which 
 A lies, shall be as near to A in magnitude as we please. 
 
 Firstly, the series increases without limit. For, since V : V, : : 
 Vj : Vg, and V and V^ are the greatest and least, we have 
 
 V -h V2 is greater than V^ + Vi 
 or V2 — Vi is greater than V^ — V 
 
 Or, Vg exceeds Vj by more than V, exceeds V. Similarly, V3 exceeds 
 Vg by more than Vg exceeds V^ ; and so on. But if to V were added 
 continually the same quantity, the result would come in time to 
 exceed any given magnitude ; still more when a greater quantity is 
 added at every step. 
 
 Secondly, since then we come at last to Vn less than A, while 
 Vn+i exceeds A, it is plain that A will not differ from either by so 
 much as they differ from each other. But because 
 
 V. : V„^i : : V : V, 
 we have V„+i-V„ : V„ : : V,-V : V 
 
 If then Vi— V be so small that m (Vi— V) shall not exceed V, 
 neither will m{Vn+i — V„) exceed V„, and of course not A . Let w 
 be any given number, however great, and let Vj — V be less than the 
 mth part of V ; then will Vn+i — Vn be less than the 7wth part of A; 
 or, by taking m sufficiently great, may be made as small as we please. 
 Whence the second part of the theorem. 
 
 Theorem. In the preceding series, the selection 
 V V„ V,„ V3„ &c. 
 
 constitutes a similar series of continued proportionals. For, since 
 any two consecutives in the upper line next given are proportional 
 to those under them in the lower, 
 
 v, V„ V, v„ 
 
 v„ V„+. V„+2 v,„ 
 
 we have (xxii.) V : V^ : : V^ : V2„ : and so on. 
 
 If between each of the terms of the series we insert the same number 
 of mean proportionals, the series thus formed will have the same pro- 
 perties as the original. Let us say we insert two mean proportionals 
 between each two terms. Then we have 
 
 H 
 
74 CONNEXION OF 
 
 V K K' Vi L L' v. M M' V3 
 
 Now the only question about the continuance of the same ratio from 
 term to term is in the ratios V, : L, Vj : M, &c. But I say that since 
 
 V : K : ; K : K' : : K' : Vi V : V^ 
 
 Vi : L : : L : L : : L' : V, ^""^ V^ : V, 
 
 that V : K : : Vj : L. For if not, let these latter ratios differ ; say 
 
 V is to K more than V, is to L. Then is K to K' more than L is 
 to L' ; and hence (presently will be shewn) the ratio compounded of 
 
 V to K and K to K', or V : K', is greater than that compounded of 
 V, : L and L : U or V : L'. Similarly, V to K' and K' to V, being 
 more than Vj : L' and L' : Vg, we have V : Vj is more than V, to Vg, 
 which is not true. Therefore V is not to K more than Vj to L ; a 
 similar process shews that it is not less : consequently, 
 
 V : K ; : Vi : L 
 
 or the continuance of the primary ratio is uninterrupted. 
 
 The theorem assumed in the above is thus proved. If A : B more 
 than P : Q we have inK greater than nB, while wP is less than nQ; 
 or any other descending assertion. And if B : C more than Q : R, 
 we have ^B greater than 3/C, while a-Q is less than y R. Or we have 
 
 mxA greater than nx^, nx^ greater than nyC, or mxk greater than nyC 
 ?w.rP less than nxQ, nx(i less than ny^, ox mxV less than wyR 
 
 that is, A is to C more than P is to R ; which is what we assumed. 
 If then we insert a mean proportional between V and A, giving 
 
 V M A 
 
 if between each we insert a mean proportional, we have 
 
 V M' M M" A 
 
 If we proceed in this way, we shall come at last to a series of the 
 form 
 
 V V, V„ V„_,(V„ = A) 
 
 in which no two quantities differ by so much as a given quantity K. 
 We can actually insert one mean proportional between any two 
 quantities ; it is done in geometry between two lines, and (page 60) 
 two magnitudes of any sort may be made (one being given) propor- 
 tional to two lines. Thus, let A, B, C, be continually proportional 
 
NUMBER AND MAGNITUDE. 75 
 
 lines, or let B be a mean proportional between A and C. Then if 
 A and C were taken proportional to (say angles) M and K, it follows 
 that if A : B : : M : L, that M, L, and K are continued proportionals, 
 by a proof of the sort given in the lemma of the last theorem. 
 Granting, then, that every two magnitudes have one mean propor- 
 tional, we may now shew that they have any number of intermediate 
 proportionals ; as follows : 
 
 We set out with 2 quantities, and the first insertion adds 1, the 
 second 2, the third 2^, the fourth 2' and the wth 2"-^ Con- 
 sequently, n complete insertions add 
 
 1 + 2 + 22 -f + 2"-i or 2" - 1 
 
 to the first 2 ; giving 2" -[- 1 in all. Now, let us suppose that 2" + 1 
 divided by p leaves a quotient g, and a remainder r which is not 
 greater than p. Consequently, we have for the whole number (V and 
 A inclusive) after n insertions, 
 
 V = pq -{-r which is also p{q + l) — (p — ^) 
 
 and p — r is also not greater than p ; and Vm=A when m=v and 
 is greater or less than A, according as m is greater or less than v. 
 If then out of the series (the proportion being continued up to 
 Vp(g+i)) we select 
 
 V Vg V23 (Vpq less than A) 
 
 V V5+1 V2 (q+i) C^p{q+i) greater than A) 
 
 We see V and Vpq, and V and ^piq+i) each with p — 1 mean pro- 
 portionals inserted between them, namely, 
 
 V« ^2q ^(P-I)q and Yq+l V2(5+i) V(p_i)(5+i) 
 
 But from Ypq to ypiq+i) there are p passages from term to term 
 of the complete series, consequently, since each passage may be made 
 by an augmentation less than K, the difference between the two may 
 be made less than p K, which call Z. Hence we have the following 
 
 Theorem. To find two magnitudes, one greater and the other less 
 than A, but differing from it by less than a given quantity Z, between 
 each of which and V, p — 1 mean proportionals shall exist, obtained 
 by continual insertion of one mean proportional, continue the in- 
 sertion until no two successive terms shall differ by so much as the 
 
76 
 
 CONNEXION OP 
 
 pth part of the quantity Z : then the quantities required and the mean 
 proportionals shall be in the set so found. 
 
 Hence it can be shewn that there are p — 1 magnitudes (whether 
 attainable or not with any given means is not the question) which 
 are mean proportionals between V and A. Let Pj, and Qp be mag- 
 nitudes, one greater and one less than A, which have such mean 
 proportionals, namely, let the following be continued proportionals, 
 
 V Pi P2 Pp_i (Pp greater than A) 
 
 V Qi Q, Qp_i (Qp less than A) 
 
 obtained by the preceding method, from which it is apparent that 
 P, is greater than Qj. Now, exactly as in page 60, if we assume 
 Xj to set out in value =sQ,, so that V : Xp more than V : A (Xp 
 
 bring the pth of the set of continued proportionals V, Xj, Xj, ) 
 
 and to change through all possible intermediate magnitudes up to 
 X,=:Pj, or V : Xp less than V : A, there is but this alternative; 
 EITHER at some intermediate point V : Xp as V : A, or Xp = A, 
 OR, there is a point at which V : Xn more than V : A, being always 
 less when X^ is greater by any magnitude however small. The latter 
 may be disproved, or the former proved, as in the page cited. 
 
 To resume the original subject. It appears, then, 1st, that if be- 
 tween V and A we continually insert mean proportionals, in such 
 manner that at every step one mean proportional is inserted between 
 every two consecutive results of the preceding step. 2d, If the series 
 be continued beyond A, preserving still the same ratio between the 
 consecutive terms of the continuation which exists between conse- 
 cutive terms lying between V and A ; then will this process leave us 
 at last with a series of consecutive proportionals, having consecutive 
 terms so near together in magnitude, that every magnitude lying be- 
 tween V and any we please to name, shall have a term of the series 
 differing from it by less than Z, however small Z may be. 
 
NUMBER AND MAGNITUDE. 
 
 77 
 
 Let us now make OK and HL perpendicular to any chosen 
 line OM, and let V be the line OK, A the line HL. Bisect OM in 
 C, and erect CD the mean proportional between OK and HL. 
 Bisect OC and CH, and erect the mean proportionals between OK 
 and CD, and between CD and HL. Continue this process, and we 
 shall thus get an increasing number of points between K and L, 
 which will soon give to the eye the idea of a curve line rising from 
 K to L. When we have thus divided OH into 2" parts, by n in- 
 sertions, giving 2" + 1 lines, we may, by setting off portions equal 
 to those intercepted in OH, continue that line on one side and the 
 other, and thus continue the scale of proportionals and the series of 
 points on one side and on the other of O and H. However far we 
 may go we can never complete this curve ; but if we admit that a 
 curve exists, wherever a series of points can be laid down, as many as 
 we please, and consecutively as near as we please, then we have 
 a right to assume this curve as existing, and, for purposes of' rea- 
 soning, as constructed. Call this the exponential curve, (exponere, to 
 set forth), which expounds ratios, a phrase to which we shall presently 
 give meaning. That the student may not suppose we are using an 
 old word in a new sense, it is necessary to inform him that this curve, 
 or rather the process which we have illustrated by it, is older than the 
 algebraical symbol a^, and that x gets the name of exponent from it. 
 We shall presently see the analogy. 
 
 The exponential curve being given, every line OG has its place 
 MP among the ordinates of the curve, and its abscissa OM, which 
 expounds or sets forth that place. From the nature of the formation, 
 it is evident that a given line has but one exponent, and that the 
 order of magnitude of lines (to the right of O), is also that of their 
 exponents. 
 
78 CONNEXION OF 
 
 And the main property of the curve is this : that a fourth propor- 
 tional to any three lines (V being one), OK, MP, M'P', may be 
 found by adding the exponents OM and OM' (making OM"), 
 and finding the line M"P" expounded by that sum. To prove this, 
 make n sets of insertions in O H, and suppose M P to lie between 
 Vm and Vm + h while M'P' lies between W and Vm'+i. Now, in 
 the series of continued proportionals, 
 
 VV, V, (VAi = A)V,V •••• 
 
 I say that V : V„. : V„, : V„^„. 
 Fof V V, V, V„.: • V„ 
 
 * m' * m'+l ^ m'+2 ^ m'+m—l * m'+m 
 
 we have V : Vj ::¥,„. : V;„.+i &c. &c. 
 
 whence V : V,„ : Y^. : Y^^„, 
 
 Similarly, V : V„,+i : V,„.+i : y„,+m'+2 
 
 Now, by a lemma we sliall presently shew, since M P lies be- 
 tween Vm and Vm + i, and M'P' lies between Vm' and Vrw'-j-i, the 
 fourth proportional required lies between Vm+m' and Vm+m'+2. Let 
 K be the value of one of the last subdivisions of OH ; then we have 
 supposed OM to lie between mK and (m-|-l)K, and OM' between 
 m'K and (m'+l)K. The preceding makes it evident that the fourth 
 proportional has an exponent between {m-\-m')K and (w -f-m' 4- 2)K ; 
 while the sum of the exponents OM and OM' also lies between 
 {m-\-m')K and (w + m'+2)K. Since K can be made as small as 
 we please, it must follow that the sum of the exponents is the ex- 
 ponent of the fourth proportional; for two different magnitudes can- 
 not lie between two quantities which can be made as near as we 
 please, as can (m -\- m')K and (m -j- m') K + 2 K. If the two approxi- 
 mating magnitudes approach to each other, keeping one of two 
 different magnitudes between them, they must, at last, leave out the 
 other. 
 
 The lemma alluded to is as follows : If 
 
 A : B::C : D 
 and A : B + B' : : C + C : D + D' 
 
 Then if A, B + X, C-|-Y, D+Z, be also proportionals, where X 
 and Y are less than B' and C, then Z must be less than D' ; for A is 
 
NUMBER AND MAGNITUDE. 79 
 
 to B + X more than A is to B + B' ; or (substituting equal ratios), 
 C + Y is to D + Z more than C + C! is to D + D'. Still more is 
 C+C (remember that C is greater than Y) to D + Z more than 
 C+C to D+D'; that is, D + Z is less than D+D', or Z less 
 than D'. 
 
 The following property we leave to the student to deduce from 
 the last. If there be any three lines, Xj Xj X3, expounding Y, Y^ Y3, 
 any lines whatsoever greater than V, then the exponent of the fourth 
 proportional is Xg+Xg — Xj. 
 
 These are all properties of algebraical exponents, or of logarithms, 
 (^Xoyuv a^i0f<,oi, numbers expounding ratios). We shall now make it 
 appear, that the line expounded by x is of the form a^. 
 
 Let the numerical symbol of V or O K be i? ; let that of H L or A 
 be a. Then, if arithmetical mean proportions be continually inserted, 
 we have 
 
 V (av)^ a 
 
 V c^v^ a^v^ cflv^ a 
 
 V a^v^ a^v^ a^v^ ah^ a 
 
 or generally, when 2" — 1 (say j9 — 1) mean proportionals are in- 
 serted between v and a, the with of these proportionals is 
 
 m - m m 
 
 (2" = p) aP V P which is ^kp 
 
 if we suppose a^=.vk. Now, let us suppose a number y thus ex- 
 pounded by X ; and after n insertions, let this number x lie between 
 ma and (m-f-l)a, a being the pih part of OH, (let OH be c). We 
 have then 
 
 lies between m- 
 P 
 
 and {m + 1)- 
 
 m 
 
 between c — 
 
 P 
 
 1 m , c 
 
 and c — + - 
 
 P P 
 
 -=o^ + /3 
 
 (^<P 
 
 Therefore, — = 
 
 pec 
 
 Consequently the number expounded by m a, or m -, or x — /3 is 
 
 Vk ' 
 
80 CONNEXION OF 
 
 and since /3 diminishes without limit as the insertions continue, the 
 
 X 
 
 number expounded by x is vk*^ . That is, if we adopt general nu- 
 merical symbols, let O M = .r, M P =y, and we have 
 
 X 
 
 or, if we let OK represent the linear unit(v=l), and let 0H = 
 OK = l, HL = 10, or /c = 10, we have 
 
 or X is the common logarithm of 3/. 
 
 From the curve we see how it is that magnitudes less than V 
 are expounded by negative quantities, with other well-known pro- 
 perties of logarithms. 
 
 We see then, that the assertion " the common logarithm of 2 is 
 •30103 very nearly," may be thus made; which is perhaps the most 
 distinct view that can be given of a numerical logarithm. If we make 
 10 V the hundred thousandth magnitude in a series of proportionals, 
 
 V, Vi, Vs, (Vioo,ooo=10) Vioo,ooi, &^- 
 
 then will the 30103rd of these proportionals, or Vg^j^g, be very nearly 
 equal to 2 V. 
 
 If we chose, we might, granting that the exponential curve can be 
 constructed, make V/cX by definition the line MP ; where X stands 
 for O M, and k for the ratio of H L to O K. From this it would 
 readily be deduced, that when k represents a commensurable ratio, 
 
 and X is — linear units where V/c^i has an arithmetical existence, 
 
 P 
 the results of this theory are the same as those of common algebra. 
 
 And from hence it appears, that the science known by the name of 
 the application of algebra to geometry (of which it is the foundation, 
 that a linear unit being given, every expression of algebra may be 
 considered as a length, or at least the symbol of the ratio of a length 
 to that unit) does, in point of fact, make this additional assumption, 
 while an application of geometry (with this assumption) to algebra, 
 would take away all want of rigorous conception of the meaning 
 of algebraical formulae, so far as the meaning of the exponent is 
 concerned. 
 
 The view above given is very nearly that by which logarithms 
 
NUMBER AND MAGNITUDE. 81 
 
 were first calculated, but the method was not so general. The natural 
 logarithms (see my Algebra, p. 226) arose thus. If we suppose a 
 very large number of mean proportionals, then V and Vj will be very 
 nearly equal. Let V^ = V + X, then if we assume O H, so that X 
 shall expound V + X when X is very small, or more correctly, if we 
 suppose the limit of (V+X) divided by the magnitude expounded by 
 X, as X diminishes without limit, to be unity, we have the first, or 
 Napier's system. 
 
APPENDIX. 
 
 ON THE DEFINITIONS, POSTULATES, AND AXIOMS 
 OF EUCLID. 
 
 I HERE propose to endeavour to make such a subdivision of the 
 definitions, &c. at the beginning of the First Book, as may enable the 
 student to review the reasoning of the whole. 
 
 I shall consider the 10th and 11th axioms as among the postulates, 
 firstly, because some old manuscripts support this change ; secondly, 
 because the older translations (from the Arabic) support it also, and 
 even place the 12th axiom in the same list; thirdly, because it is 
 utterly impossible to place them in Euclid's list of common notions. 
 For he uses no such word as axiom (Greek though it be), but calls 
 " the whole is greater than its part," koivh hvota, that which is in the 
 conceptions of every one. Now, what is the probability that he 
 considered " all right angles are equal," as a truth familiar to the 
 understanding of every beginner in geometry? His postulates 
 {a,iTYifjt.a,roe., demands) do, according to the etymology of the word, 
 include those axioms, if not the 12th also. 
 
 1 also place out of view the axioms which belong to all kinds of 
 magnitude as much as to space, namely, from the 1st to the 9th in- 
 clusive. There remains then in the shape of limitation, or assumption, 
 six postulates, namely, three which I will call restrictive, being those 
 commonly called postulates,* and three assumptions, being the 10th, 
 11th, and 12th axioms, so called. 
 
 Some of the definitions contain assumptions of certain conceptions 
 existing to which names are to be given ; namely, those of a point, a 
 
 • I have seen the word postulate defined as a self-evident problem; 
 and axiom as a self-evident theorem. This definition is derived from 
 the character of the postulates and axioms as usually given j but from no 
 other source. 
 
84 APPENDIX. 
 
 line, the extremities of a line, a straight line, a surface, the extremities 
 of a surface, a plane surface, a plane angle, a plane rectilineal angle. 
 Others assume the possibility of certain relations existing, as will 
 appear from the form in which they are put. I shall now give the 
 definitions, classified with the corresponding postulates, in the manner 
 which appears to me to be most systematic, and placing in [ ] such 
 additions as seem requisite. 
 
 1. A point; an indefinable notion; but two persons, whatever 
 their idea of it may be, can reason together in geometry who deny a 
 point all parts or magnitude. Let it be granted that a point has no 
 parts or magnitude, and that we are concerned with no other property 
 of it, if there be any. 
 
 2. Aline; also indefinable, but those whose ideas of it allow it 
 length, and deny it breadth, can proceed. Let it be granted that all 
 reasoning upon lines is to be founded only upon the assumption that 
 they have length without breadth. [Thickness should have been 
 added, but breadth may mean breadth in any direction.] 
 
 3. The extremities of a line are points. [If this define any term, 
 it must be the term extremities^ for the other two have been defined. 
 To me it appears something like a theorem, as follows : That which 
 ends a line cannot have length, for it would be a part of the line ; it 
 cannot have breadth or thickness, which a line has not ; it has there- 
 fore the only qualities of a point on which we reason, or comes within 
 the definition of a point,] 
 
 4. A straight line; an indefinable notion, except by the rough 
 idea that it does not go on one side or the other of the two points, 
 [which is no definition, because it assumes the thing in question.] 
 Let it be granted, as a common notion, that two straight lines do not 
 enclose a space, or have not two points in common, without having 
 all intermediate points in common. Whatever the idea of a straight 
 line may be, this is the only property which will be appealed to. 
 
 5. Surface; an indefinable notion; those whose ideas give it 
 leno^th and breadth, but deny it thickness, have the means of reasoning 
 upon it in geometry. 
 
 6. The extremities of a surface are lines. (See remarks on 3.) 
 
 7. A plane surface; an obvious notion, roughly defined by lying 
 evenly between bounding straight lines. [This notion, however 
 obvious, does admit of a stricter definition. It is a surface of such 
 kind that ani/ two points in it being joined by a straight line, all 
 
APPENDIX. . 85 
 
 intermediate points of the straight line are on the surface. This 
 property is tacitly appealed to throughout.] 
 
 8. A plane angle ; the inclination, or bending towards each other 
 of two lines in a plane. [This definition is superseded by the next; 
 no angle except one made by straight lines is ever used.] 
 
 9. A rectilineal angle (plane), the inclination of two straight lines. 
 [An obvious notion of opening; it is tacitly assumed that we know 
 how to determine when two angles are equal, or when one of them 
 exceeds the other, as in the fourth proposition.] 
 
 10. Right angles are those made by a straight line, called a 
 perpendicular, which falls on another straight line, making equal 
 angles on both sides. Postulate ; let it be granted that all right angles 
 are equal. [This is far from an obvious postulate ; the reason for it 
 seems to have been as follows : That two straight lines which coincide 
 in two points coincide in every point between them, has been admitted ; 
 it is sufficiently obvious to sense that they coincide beyond or on each 
 side of the two common points ; that is, they coincide altogether, 
 throughout all possible length. This seems an infinite assumption ; 
 and if it be assumed instead that all right angles are equal, it may be 
 proved afterwards that no two straight lines have a common segment ; 
 that is, that two straight lines which coincide for any length, never 
 afterwards separate. But it may be shewn, that the assumption of all 
 right angles being equal, amounts to the same infinity of assumption ; 
 as follows : The right angle is by definition the half of the opening 
 which two straight lines make, when one is the continuation of the 
 other, as AB, BC. To assume that all right angles are equal, is to 
 assume that the doubles 
 
 h -r -\ 
 
 ABC 
 
 V -r H 
 
 D E F 
 
 of right angles are equal ; that is, that if we lay B on E, with E D 
 coinciding with BA, then EF and BC will coincide. Now it is 
 precisely the same thing to assume, that when AB is made to coincide 
 with D E up to the point E, that the two coincide beyond it. 
 
 I should recommend the student to make to the assumption that 
 two straight lines cannot coincide in two points without coinciding 
 between them, the addition that they also must coincide beyond them. 
 
86 . APPENDIX. 
 
 It may then be directly proved that all doubles of right angles are 
 equal, and thence that all right angles are equal.] 
 
 The definitions 11, 12, 13, 14, need no remark, being purely 
 nominal. 
 
 14. The circle f a plane figure, having all points of its boundary 
 (15. the circumference) equally distant from a given point (16. the 
 centre) within it. [Here is tacitly a postulate, namely, that this point 
 lies within the figure. It is also assumed in the first proposition, that 
 if any point of a circle be within another, the two circles must 
 intersect. There are several assumptions of this kind, which shew 
 that Euclid did not affect that extreme form of accuracy which sub- 
 sequent commentators have attributed to him. The assumption of a 
 circle assumes the existence of an isosceles triangle.] 
 
 17. A diameter of a circle is a line passing through the centre, 
 and terminated both ways by the circumference ; it divides the circle 
 into two equal parts, or (18. semicircles). [Here is a demonstrable 
 theorem positively assumed. The application of one part of the 
 circle to the other (as by revolution of one-half round the diameter) 
 as in the fourth proposition, would prove it.] 
 
 From (19.) to (23.), the definitions are merely nominal. 
 
 24. If there be a triangle having three equal sides, let it be called 
 equilateral. [In this form I give all definitions, the existence of the 
 objects of which is to be established.] 
 
 25. An isosceles triangle is one having two sides equal. 
 
 26. A scalene triangle has the three sides unequal.] This defini- 
 tion is never used.] 
 
 (27.) and (28.) are nominal ; (29.) tacitly refers to the thirty- 
 second proposition ; and from (30.) to (33.), should be written in 
 the manner of (24.) 
 
 (35.) If there be two right lines, which being produced ever so 
 far on the same side never meet, let them be called parallels. And 
 let it be granted, that if two right lines falling upon a third make 
 interior angles together less than two right angles, they are not 
 parallels. [This bone of contention, when reduced to the form in 
 which it is most palpable to the senses, is as follows : Let it be granted 
 that two right lines which meet in a point, are not both parallel to any 
 third line. This assumed, Euclid's axiom follows. For he is able to 
 shew that the one parallel which he afterwards draws, through a point 
 to a given line, has the property of making the two internal angles 
 
APPENDIX. 
 
 87 
 
 equal to right angles : there is but one parallel; consequently all lines 
 which have not that property are not parallels.] 
 
 It remains to add what I have called the restrictive postulates. 
 I cannot believe that Euclid, who appears to assume vert/ obvious 
 propositions, even when he might prove them, could have intended to 
 require formally the admissions that a straight line may join two 
 points, and may be continued, and that a circle may be drawn with a 
 given centre and radius. If this had been the case, why not assume 
 (Prop. IV.) that two straight lines may be drawn making equal angles 
 with two other straight lines, — a conception more difficult than that 
 a straight line may be drawn. I conceive, therefore, that the meaning 
 of the three assertions commonly called postulates, is as follows: 
 Let it be considered as intended, that no assumption of processes shall 
 be made, except only the drawing of a straight line between two given 
 points, the continuation of any terminated straight line to any in- 
 definite (not given) distance, and the construction of a circle with a 
 given centre and radius. 
 
 THE END. 
 
 LONDON i — PRINTED BY JAMES MOYES, 
 Ca&tle Street, Leicester Square. 
 
[LIST, No. 3.] 
 
 Upper Gower Street ^ March 1, 1835, 
 
 WORKS 
 
 MATHEMATICS, NATURAL PHILOSOPHY, 
 
 HISTORY, &c. 
 
 PRINTED FOR TAYLOR AND WALTON, 
 
 BOOKSELLERS AND PUBLISHERS TO THE UNIVERSITY OF LONDON. 
 
 *»* Some €»/ the Books in the following List are in use at the University or Us School. 
 
 THE ELEMENTS OF ARITHMETIC. 
 
 By Augustus De Morgan, Esq. Third Edition, royal r2mo, 45, cloth. 
 
 " Since the publication of the first 
 edition of this work, though its sale has 
 sufficiently convinced me that there exists 
 a disposition to introduce the principles 
 of arithmetic into schools, as well as the 
 practice, I have often heard it remarked 
 that it was a hard book for children. I 
 never dared to suppose it would be other- 
 wise. All who have been engaged in 
 the education of youth are aware that it is 
 a hard thing to make them think; so hard, 
 indeed, that masters had, within the last 
 few years, almost universally abandoned 
 the attempt, andtaughtthem rulesinstead 
 of principles ; by authority, instead of 
 demonstration. This system is now pass - 
 ir)g away ; and many preceptors may be 
 found who are of opinion that, whatever 
 may be the additional trouble to them- 
 selves, their pupils should always be 
 induced to reflect upon, and know the 
 reason of, what tliey are doing Such I 
 would advise not to be discouraged by the 
 failure of a first attempt to make the 
 learner understand the principle of a rule. 
 It is no exaggeration to say, that, under 
 the present system, five years of a boy's 
 life are partially spent in merely learning 
 the rules contained in this treatise, and 
 those, for the most part, in so imperfect 
 
 a way, that he is not fit to encounter any 
 question unless he sees the head of the 
 book under which it falls. On a very 
 moderate computation of the time thus 
 bestowed, the pupil would be in no respect 
 worse oflF, though he spent five hours on 
 every page of this work. The method of 
 proceeding which I should recommend, 
 would be as follows: — Let the pupils be 
 taught in classes, the master explaining 
 the article as it stands in the work. Let 
 the former, then, try the demonstration 
 on some other numbers proposed by the 
 master, which should be as simple as 
 possible. The very words of the book 
 may be used, the figures being changed, 
 and it will rarely be found that a learner 
 is capable of making the proper alter- 
 ations, without understanding the reason. 
 The experience of the master will suggest 
 to him various methods of trying this 
 point. When the principle has been thus 
 discussed, let the rule be distinctly stated 
 by the master, or some of the more intel- 
 ligent of the pupils ; and let some very 
 simple example be worked at length. The 
 pupils may then be dismissed, to try the 
 more complicated exercises with which 
 the work will furnish them, or any others 
 which may be proposed." — Preface. 
 
 THE ELEMENTS OF ALGEBRA, 
 
 Preliminary to the Differential Calculus, and fit for the higher classes of Schools 
 in whicli the Principles of Arithmetic are taught. By Augustus De Morgan, Esq. 
 Royal r2mo. 9s. cloth. 
 
 III. 
 ARITHMETICAL ALGEBRA. 
 
 By G. J. P. "White, Professor of Mathematics in the University of London. 
 One vol. 8vo. (Preparing.) 
 
 LESSONS ON NUMBER, 
 
 As given at a Pcstalozzian School at Cheam, Surrey. Second Edition. 
 The Master's Manual. 12mo. 4s. 6rf. cloth. ^ /o 1 1 t 1 ' ^ 
 
 The Scholar's Praxis. 12mo. 2*. bound. 5 
 
 A TREATISE ON ARITHMETIC, 
 
 Theoretical and Practical. By the Rev. Dionysius Lardner, LL.D. 
 Foolscap Octavo, Qs. cloth. 
 
WORKS ON MATHEMATICS, HISTORY, ETC. 
 
 CONVERSATIONS ON ARITHMETIC, 
 
 Adapted for the Use of Preparatory Schools and for Domestic Instruction. 
 Part I. Foolscap 8vo, 3s. cloth. 
 
 VII. 
 
 A SYSTEM OF ARITHMETIC. 
 
 Practical and Theoretical. By the Rev. William Ritchie, LL.D., Professor of 
 Natural Philosophy and Astronomy in the University of London. 12mo, 3s. hound. 
 
 PRINCIPLES OF GEOMETRY, 
 
 Familiarly illustrated, and applied to a variety of useful purposes. Designed 
 for the Instruction of young Persons. By the Rev. Professor Ritchie, LL.D., 
 F.R.S. 12mo, with 150 Wood Cuts. 3s. 6c?. cloth. 
 
 " The practical applications which are added, must render the study very de- 
 lightful to the young, since the Exercises on the Principles will be found as amusing 
 as the ordinary sports of chiUlhoQd." — Athenceum, Sept. 28, 1833. 
 
 " Dr. Ritchie's little elementary work is excellently well adapted to its object. It is 
 brief, plain, and full of all that is necessary ; curious and useful in its application, and 
 beyond any other of the kind now existent, in its familiar and distinct explanation of 
 some of the instruments required in the practical application of the principles laid 
 down and demonstrated." — Spectator, Sept. 7, 1833. 
 
 AN INSTRUMENT FOR TEACHING GEOMETRY, 
 
 Convertible into a Theodolite, Spirit Level, Hadley's Sextant, and WoUaston's 
 Goniometer, has heen prepared, aud maybe had of the publisher. Price 21. 12s. 6d. 
 
 PRINCIPLES OF THE DIFFERENTIAL AND INTEGRAL 
 CALCULUS, 
 
 Familiarly Illustrated, and applied to a variety of Useful Purposes. Designed 
 for the Instruction of Young Persons. By the Rev. Professor Ritchie, LL.D., 
 F.R.S. With Wood Cuts. 12mo, 4s. 6d. cloth. 
 
 X. 
 
 CONVERSATIONS ON GEpMETRY, 
 
 Adapted for the Use of Preparatory Schools and Domestic Instruction. 
 Foolscap 8yo, 3s. cloth. 
 
 THE ELEMENTS OF EUCLID, 
 
 With a Commentary and Geometrical Exercises. By the Rev. Dionysius 
 Lardner, LL.D. Fourth edition, 8vo, 7s. boards. 
 
 AN ANALYTICAL TREATISE ON PLANE AND 
 
 SPHERICAL TRIGONOMETRY. 
 
 By the Rev. Dionysius Lardnkr, LL.D. Second Edition. CoiTected and improved. 
 
 '8vo, 12s. cloth. 
 
 XIII. 
 
 AN ELEMENTARY TREATISE ON THE DIFFERENTIAL 
 AND INTEGRAL CALCULUS. 
 
 By the Rev. Dionysius Laroner, LL.D. 8vo, 21s. 
 
PRINTED FOR TAYLOR AND WALTON, UPPER GOWER STREET. 
 XIV. 
 
 BARLEY'S SCIENTIFIC LIBRARY, 
 
 FOR THE USE OF SCHOOLS, PRIVATE STUDENTS, ARTISTS, AND MECHANICS. 
 
 It is the purpose of this "Work to furnish a Series of Elementary Treatises on 
 Mathematical Science, adapted to the wants of the public at large. To youth of 
 either sex at public and private schools ; to persons whose education has been neg- 
 lected, or whose attention has not been directed in early life to such studies; and to 
 Artists and Mechanics, these little works will be feund particularly suited. The 
 principles of the various Sciences are rendered as familiar and brought as near to 
 our commonest ideas as possible ; the demonstrations of propositions are made plain 
 for the mind, and brief for the memory ; and the Elements of each Science are 
 reduced not only to their simplest but to their shortest form. 
 
 1. A SYSTEM OF POPULAR GEOMETRY. 
 
 Containing in a few Lessons so much of the Elements of Euclid as is necessary 
 and sufficient for a right understanding of every Art and Science in its leading 
 Truths and general Principles. By George Darley, A.B. Third Edition. Price 
 4s. 6d. cloth. 
 
 i I 2. COMPANION TO THE POPULAR GEOMETRY. 
 
 In which the Elements of Abstract Science are familiarised, illustrated, and ren- 
 dered practically useful to the various purposes of Life, with numerous Cuts. Price 
 4s. 6d. cloth. 
 
 3. A SYSTEM OF POPULAR ALGEBRA, 
 
 With a Section on Proportions and Progressions. Second Edition. Price 45. 6d. 
 
 cloth. 
 
 4. A SYSTEM OF POPULAR TRIGONOMETRY, 
 
 Botli Plane and Spherical, with Popular Treatises on Logarithms, and the 
 Application of Algebra to Geometry, Second Edition. Price 3s. 6d. cloth. 
 
 •' For Students who only seek this limited knowledge of these Sciences, there are 
 perhaps no Treatises which can be read with more advantage than Darley's Popular 
 Geometry aad Algebra." — Library of Useful Knowledge, Article "Mechanics." 
 
 Katural pi&ilosoplDB. 
 
 THE STEAM ENGINE, 
 
 Familiarly Explained -and illustrated ; with its application to the Arts and Ma- 
 nufactures, to Steam Navigation, and Rail Roads ; with Plain Maxims for the 
 Guidance of Railway Speculators. By the Rev. Dionvsius Lardner, LL.D. Fifth 
 Edition. One vol. 12mo. Illustrated with Engravings and Wood Cuts. 7s. 6d. 
 cloth. 
 
 II. 
 
 FAMILIAR ASTRONOMY. 
 
 By George Darley, A.B., Author of a System of Popular Geometry, &c. 
 12mo, with Engravings, 5s. cloth lettered. 
 
 " There is a vast deal of astronomical information conveyed in a most winning and 
 unassuming manner in this delightful little volume, which, not less for the novelty of 
 its plan, tlian the extent of its intelligence, reflects infinite credit on the taste and 
 talents of its projector and editor, Mr. Darley." — Sun, April 5, 1830. 
 
WORKS ON MATHEMATICS, HISTORY, ETC. 
 
 ON THE ORIGIN OF UNIVERSITIES AND ACADEMICAL 
 DEGREES. 
 
 By Hkkry Malden, M.A., late Fellow of Trinity College, Cambridge; Professor 
 of Greek in tlie University of London. Foolscap 8vo, 35, 6d. cloth. 
 
 NIEBUHR'S HISTORY OF ROME. Vol. I. 
 Translated by Julius C. Hare, M.A., and Connop Thirlwall, M.A., Fellows of 
 Trinity College, Cambridge. 
 Second Edition, revised, Avith the Corrections and Additions made in the Third 
 Edition of the Original, 8vo, I6s. 
 
 NIEBUHR'S HISTORY OF ROME. Vol. II. 
 
 Translated by Julius C. Hare, M.A., and Connop Thirlwall, M.A., Fellows 
 
 of Trinity College, Cambridge. 8vo, 16s. 
 
 " Niebuhr undertook to write the His- 
 tory of Rome from the earliest ages of the 
 city to the establishment of the empire of 
 Augustus. Of this great work he accom • 
 plished only a portion ; and his History 
 will remain to succeeding ages as a frag- 
 ment, but it is a fragment which may be 
 compared to the unfinished colossal statues 
 that are found lying in the granite quarries 
 of Syene, conceived with all the vastness 
 and precision of Egyptian art, which, had 
 they been finished, might have overtopped 
 the gigantic Memnon, but which, when 
 they were relinquished by the hand that 
 first fashioned them, were destined to 
 remain for ever imperfect. We should as 
 soon expect an artist to arise, who should 
 elaborate, and erect amid the Theban 
 
 temples, a half hewn Rameses, as a scholar 
 who should complete the Roman- History 
 of Niebuhr." — 
 
 " Here we close our remarks upon this 
 memorable work; a work which, of all 
 that have appeared in our age, is the best 
 fitted to excite men of learning to intel- 
 lectual activity; from which the most 
 accomplished scholar may gather fresh 
 stores of knowledge, to which the most 
 experienced politician may resort for 
 theoretical and practical instruction, and 
 which no person can read as it ought to 
 be read, without feeling the better and 
 more generous sentiments of his common 
 human nature enlivened and strength- 
 ened." — Edinburgh Review, Jan. 1833. 
 
 A VINDICATION OF NIEBUHR'S HISTORY OF ROME 
 
 From the Charges of the Quarterly Review. By Julius Charles Hare, M.A., 
 Fellow of Trinity College, Cambridge. 8vo, 2*. 6d. 
 
 BRIEF OUTLINES OF DESCRIPTIVE GEOGRAPHY, 
 
 To which is subjoined a Table of Latitudes and Longitudes. By Henry H. 
 Davis. Foolscap 8vo, with Maps, 2s. 6d. cloth. 
 
 " The Outlines here presented to the Public, are the results of actual extempora- 
 neous Lessons." — Preface. 
 
 geography"simplified ; 
 
 Being a brief Summary of the principal Features of the Four Great Divisions of 
 the Earth, with a more detailed Account of the British Empire. Also, a familiar 
 Explanation of the Use of the Globes, and an Appendix. By an Experienced 
 Teacher. 12mo, is. bound. 
 
 III. 
 
 OUTLINE MAPS. 
 
 Mercator Europe. — British Isles. 
 
 Three Maps, folio, stitched in cover, Is. Single Maps, 4rf. each. 
 
 GEOGRAPHICAL PROJECTIONS. 
 
 Mercator. — Europe. — British Isles. 
 Three Maps, folio, stitched in cover. Is. Single Maps, id. each. 
 
 ^.-A 
 
]^rf%*' 
 
 'Li,\ -i^. J%!\M£i^CjSi ^I^SS 
 
 U DAY US:e 
 
 RETURN TO DESK FROM WHICH BORROWED 
 LOAN DEPT. 
 
 This book is due on the last date stamped below, or 
 
 on the date to which renewed. 
 
 Renewed books are subject to immediate recall. 
 
 89}iov'&7fiXX 
 
 . M 17 2002 
 
 f^ETC'D LD 
 
 ^AN 9 t qsa 
 
 20Feb'59MR 
 
 REC^P LD 
 
 m^tt^RYiuiea 
 
 NOV^i 1952 
 
 m\^ 
 
 Tm^ 
 
 f^/^ 
 
 3i0 2001 
 
 o£c 1 7 ^mi 
 
 LD 21A-50m-8.'57 
 (C8481sl0)476B 
 
 General Library 
 
 University of California 
 
 Berkeley 
 
f 
 
 I 
 
• ,>^ ;•*"!-