Feb. 22, 1877] 



NATURE 



353 



They are, moreover, being issued in such numbers, under 

 the present demand for popular education, that their very 

 likeness to one another is fatiguing. They require also in 

 their construction the rare faculty, whether intuitive or 

 gained by long experience, of insight into a student's 

 probable difficulties ; for it seems desirable that they 

 should rather aim at being employed as condensors and 

 systematisers of knowledge already acquired generally 

 from the study of larger and more diffuse treatises, than 

 as independent works. It is in this respect that useful 

 practical knowledge differs from "cram"; a distinction 

 very real, though more difficult to define than to under- 

 stand. The concentrated food offered by such compila- 

 tions is less easy of digestion, and more readily expelled 

 from the mental economy, than that which is more 

 gradually administered and more completely assimilated. 



The writer of the present manual has, for instance, 

 only seventy pages to devote to Sound, one hundred and 

 eighteen to Light, and ninety-one to Heat, exclusive of the 

 Appendix. But it is remarkable how much he succeeds in 

 compressing within these very restricted limits. The 

 illustrative experiments are, as a rule, simple and well 

 chosen, though occasionally trite, and even of doubtful 

 accuracy ; as is seen in the drawing of the periodic curve 

 of a musical sound at p. 40, and that of dispersion of 

 light on p. 135. On the other hand, the use of a long 

 spiral steel spring to illustrate waves of compression and 

 rarefaction, the description of the effects of Temperature 

 on Sound-waves, and the chapters on Interference, Dif- 

 fraction, and Polarisation of Light, especially in its 

 Circular and Rotatory forms, are ingenious and easy 

 to comprehend. 



A few simple numerical examples are given of each 

 important law, with their solutions, and the mode of 

 working out ; a method which probably tends more than 

 any other to fi.x essential points on the memory of the 

 student. W. H. Stone 



LETTERS TO THE EDITOR 



[The Editor does not hold himself responsible for opinions expressed 

 by his correspondents. Neither can he undertake to return, 

 or to correspond with the writers of, rejected manuscripts. 

 No notice is taken of anonymous communications i\ 



Postulates and Axioms 



A STRONG committee, appointed, or rather re-appointed, for 

 the purpose, reported last year to the British Association upon 

 the Syllabus drawn up by the Association for the Improvement 

 of Geometrical Teaching. I have only just seen a copy of the 

 report, and I wish to point out that it incidentally touches in a 

 misleading fashion upon a matter which, though primarily of only 

 historical interest, is really of theoretical importance too, if not 

 (in the strictest sense) for the special purpose of the committee ; 

 I mean upon the different ways of distributing the fundamental 

 a<-sumptions under the two heads of postulate and axiom. 



Let us stop for a moment at the historical point of view. It 

 is well known that the received text of Euclid, which we may 

 consider represented by David Gregory's edition (Oxford, 1703), 

 misplaces the assumption about right angles, the assumption at 

 the base of the theory of parallels, and the assumption that two 

 straight lines do not inclose a space. That is to say, whereas 

 n the correct text these are the 4tb, 5th, and 6th postulates, 

 tie received text inakes them the loth, nth, and 12th common 

 lotion?, or, as we usually say, axioms. 



Now, when the report speaks of Euclid in this connection, it 

 neans something nearly identical with the received text. Not 

 luite, however ; for, though the language is not clear in all 

 cspect?, it clearly says thus much, that Euclid divided the axioms 

 i to general and specially geometrical. But this is not the case 

 ii either text ; for in both texts the first seven common notions 

 ae general, the 8th geometrical, and the 9th general again, nor 

 is the 8th distinguished from the rest by its grammatical form. 

 Bit whether you follow the received text or depart from both, 

 it is unhistorical to affirm of Euclid what is not true of the cor- 

 ret text. 



Let us now consider the theoretical significance of the two dis- 



tributions. The case is thus stated by De Morgan, under 

 Eucleides, in Smith's " Dictionary of Greek and Roman Bio- 

 graphy," p. 66b : — " The intention of Euclid seems to have been 

 to distinguish between that which his reader must grant, or seek 

 another system, whatever may be his opinion as to the propriety 

 of the assumption, and that which there is no question everyone 

 will grant The modern editor merely distinguishes the assumed 

 problem (or construction) from the assumed theorem'^ This latter 

 distinction is at least as old as Proculus ; but to De Morgan 

 it is Euclid's, at least as concerns right angles and parallels, 

 that "seems most reasonable ; for it is certain," he continues, 

 "that the first two assumptions can have no claim to rank 

 among common notions or to be placed in the same list with 

 * the whole is greater than its part.' " We need not pursue the 

 modern editor's distinction further ; but Euclid's acquires a more 

 definite significance in relation to those generalised conceptions 

 of space which, since De Morgan wrote these words, have almost 

 passed into popular science. This in its generality is a difficult 

 subject, but for the present purpose it is enough to regard plane 

 geometry as a particular case of the geometry of points and lines 

 on a given surface. 



In this view the postulates specify the attributes of the plane 

 which make plane geometry what it is. Thus the first three, 

 whatever else they do, provide that the power of drawing dia- 

 grams shall not be restricted by boundaries, and the fourth, " all 

 right angles are equal," affirms that a complete rotation is the 

 same in quantity at all points ; thereby the first three exclude 

 surfaces having such a singular locus as a cuspidal line, and the 

 fourth excludes surfaces having such a point as the vertex of a 

 cone. Again the fifth excludes anticlastic surfaces, and the sixth 

 synclastic ones and any which, like the common cylinder, returns 

 into itself. Nothing remains but the plane and such developable 

 surfaces as the parabolic cylinder to which mutatis mutandis 

 everything in plane geometry will equally apply. 



The axioms, on the contrary, specify r.o property of any 

 class of surfaces. This is crucially instanced in the one axiom 

 (the 8th, that things congruent are equal) which does concern 

 figures traced on surfaces of only a limited class. For this 

 axiom merely says that if things coincide they are equal, not that 

 figures m different places may be brought to coincide. 



The question may be asked whether this last assumption ought 

 not to be premised somewhere ; that is, whether the method of 

 superposition ought not to have been vindicated by expressly 

 assuming that any plane figure may be laid down on any plane 

 so as to coincide with a portion of it. The omission is an ex- 

 tremely curious fact — in Euclid, I mean, for it is not at all re- 

 markable in his successors. On the one hand, express statement 

 is superfluous in the sense that the assumption is implied in the 

 last two postulates ; for the fifth affirms that the " measure of 

 curvature " of the plane is not negative, and the sixth that it is 

 not positive ; between them it is naught, and therefore constant ; 

 but this is the condition of superposablenes?. On the other hand, 

 express statement is indispensable in the sense that the student 

 cannot do without it, because the theory of measure of curvature 

 does not belong to elementary geometry. 



The fact is that Euclid has drawn the line with what is realiy 

 remarkable accuracy, but is only seen to be so in virtue of prin- 

 ciples not discerned, I beUeve, by any one before Gauss. What- 

 ever may be the explanation of this phenomenon, to ignore it in 

 speaking of Euclid's postulates and Euclid's axioms is to depart 

 from history where adherence to history would be instructive in 

 theory too. 



It is of course another question whether this distinction of 

 Euclid's ought to be preserved in books intended to supersede 

 Euclid. C. J. Monro 



Hadley, Barnet 



Just Intonation 



That Mr. Chappell misunderstands me is due parUy*to his 

 confounding vibration numbers with their ratios. Thus f J is the 

 vibration number of the supertonic, where 2 is that of the tonic ; 

 whili 524288 is not the vibration number of any musical sound, 

 though the ratio 524288 : 531441 = 2^" : -^'^ expresses an interval 

 that may be picked out fourteen times in each octave of Mr. 

 Culm Brown's keyboard. A still more complex interval 2^^ : "^^ 

 is fouiid seven times in each octave. 



I '.^.e followed Mr. Chappell's advice and purchased his six- 

 penny pamphlet, and having read it with the care it deserves, I 

 can only say I dissent from a great part of it, especially where 



