Dec. 6, 1888] 



NA TURE 



125 



fancy or an odd quotation. We read also the appen- 

 dixes, which are of a similar character with the preface ; 

 but then there remained the inner layer of the sandwich, 

 which called for a most careful overhauling. Anyone who 

 has read Mr. Dodgson's " Euclid and his Modern Rivals " 

 (cited in our present notice as " E. and R."), is prepared 

 to find close logical reasoning and acute remarks in any 

 work he undertakes, and the reader of this booklet will not 

 be disappointed in these particulars. 



Euclid's Twelfth Axiom the author asserts to be «fl/ ax- 

 iomatic, i.e. he has not met with any "bimanous biped" who 

 accepts it as intuitive truth. His quest is to find a better 

 axiom. At first sight— and the illustrative figure meets 

 us on the cover and in other places — one is disposed 

 to grant the truth of the Dodgsonian axiom, viz. that " in 

 any circle, the inscribed regular hexagon is greater than 

 any one of the segments that lie outside it." But the 

 author is not restricted to a paltry once; he can equally 

 well grapple with the problem, if his reader will 

 grant that twice, four times, eight times, or, in mathe- 

 matical parlance, 2" times, the hexagon is greater 

 than any one of the above-named segments. The 

 principal part of the text consists of five definitions, 

 six axioms, and seventeen propositions. Appendix I. 

 contains alternative proofs of certain propositions 

 consequent upon the general form of the author's axiom. 

 The " distance " between two points figures as a defini- 

 tion ("the length of the shortest path between them") 

 and as an axiom (" the length of the straight line joining 

 them"). Prop. IV. is headed a " Theorem " ; we should 

 have thought it was a " Problem." It is : — " Given a 

 triangle : to describe an equilateral triangle which shall 

 "enclose it" (third line from end — for AE read DE). 

 Prop. V. is an important one ; it runs thus : — " Given a 

 certain angle ; and given that every isosceles triangle, 

 whose vertical angle is not greater than the given angle, 

 has its base not greater than either of its sides ; to de- 

 scribe, on a given base, an isosceles triangle having each 

 base-angle equal to the given angle " (line 5 from end, 

 for " DE produced" read "BE produced"). In the 

 corollary to this, the argument seems to us to be 

 scamped : a hasty reader might think that Mr. Dodgson 

 had asswned Euc. I. 32. We would close thus : — "Angle 

 AFB is greater than angle .A.CB, .'. greater than angle 

 ABC, and a fortiori greater than angle ABF." Prop. VI. 

 is all right, but how are the figures to be constructed if 

 « > 2 ? The sum of the angles of a triangle is called, by 

 Mr. Dodgson, its "amount." In Prop. VII. (interchange 

 E and F in dexter figure) he shows that, " if a, ^ be two 

 'possible amounts'— that is, ' amounts ' belonging to ex- 

 isting triangles— then every * amount ' intermediate to 

 a and iS is also ' possible.'" Prop. VIII. shows that there 

 is a triangle whose angles are together not greater than 

 two right angles (line 7 up, p. 17, for " = " read " > " ; 

 line I up, p. 18, for " ABD " read" ADB"). At this point 

 comes the axiom, and Prop. IX. follows :— " An isosceles 

 triangle, whose vertical angle is one-twelfth of a right 

 angle, has its base less than either of its sides" (the 

 corollary applies Prop. V. above). Prop. X. shows that 

 "the angles of every equilateral triangle are together 

 not less than one-fourth of a right angle." Prop. XI. 

 establishes that " there is a triangle whose angles are 

 together not less than two right angles." This is a long 



proposition. Props. VIII. and XI. are sound on the 

 hypothesis which Mr. Dodgson seems tacitly to have 

 adopted, viz. that the " amounts " of triangles are all in 

 the same boat, either all greater than two right angles or 

 all less than the same quantity. Surely it is, a priori^ 

 conceivable that the "amounts" may be variable, and 

 then how will his proofs hold? Prop. XII. readily 

 deduces, from the previous reasoning, that " there 

 is a triangle whose angles are together equal to 

 two right angles." Prop. XIII. proves that there is a 

 quadrilateral figure with all its angles right angles {i.e. 

 a rectangle) ; and Prop. XIV. shows that the opposite 

 sides of such a figure are equal. But, " tell it not in 

 Gath," Mr. Dodgson takes up the rectangle — we repeat 

 it, takes up the rectangle — and reverses it, " so that A, B, 

 may change places." We do not object, but how about 

 the Irish bull (E. and R., p. 47) ? But this is not the 

 only surprise in store, for in Prop. XV., to prove that 

 " there is a pair of lines, each of which is ' equidistant ' 

 from the other — that is, is such that all points on it are 

 equally distant from the other line " — he makes a rectangle 

 slide along on a straight line ! Prop. XVI. proves that 

 " the angles of every triangle are together equal to two 

 right angles " ; and Prop. XVII. winds up the story with 

 showing " that a pair of lines, which are equally inclined 

 to a certain transversal, are so to any transversal." 



The appendixes repay perusal. Appendix II. dis- 

 cusses Euclid's axiom, and argues that, " though true in 

 the sense in which Euclid meant it, it is not true in the 

 sense in which we take it." In fact, Mr. Dodgson con- 

 tends that Euclid " excludes from his view both infinities 

 and infinitesimals, and considersyfw/^ magnitudes only." 

 This is rightly founded on Euclid's Book X. Prop. I. We 

 cannot go into the matter further here, but commend this 

 and Appendix III. (" How should Parallels be Defined.?") 

 and Appendix IV, (" How the Question stands To-day ") 

 to any reader who is interested in this crucial question. 

 We can quite sympathize with the author, as in times 

 past we have more than once done our little best in the 

 same direction, when he recounts how, more than once, 

 he, too, has " with clasped hands gazed after the retreat- 

 ing meteor, and murmured, ' Beautiful star, that art so 

 near and yet so far.'" To conclude, Mr. Dodgson is 

 " inclined to believe that, if ever Euc. I. 32 is proved 

 without a new axiom, it will be by some new and ampler 

 definitionof ///^r/'^/zZ/zV/^— some definition which shall con- 

 note that peculiar and mysterious property, which it must 

 somehow possess, which causes Euc, I. 32 to be true. 

 Try that track, my gentle reader ; it is not much trodden 

 as yet ; and may success attend your search ! " 



OUR BOOK SHELF. 



Primer of Micro- Petrolos^y. By W. Mawer, F.G.S. 



(London : Office of Life-Lore, 1888 ) 

 The task of introducing the student to any particular 

 branch of science requires such selective judgment, such 

 tact both in saying and in leaving things unsaid, that it is 

 not surprising if many so-called "primers" fall short of the 

 good intentions of their authors. Mr. Mawer, in this 

 little book, presupposes " an acquaintance with the 

 phenomena of pleochroism and the polarization of light," 

 and hence refers only in the briefest manner to the 

 methods employed in the examination of thin mineral 



