Supplement to '' Naiiwe," October 19, 1905 



not what Mr. Wilson does, and the consequences of 

 his procedure are very instructive, especiallv from the 

 point of view of absolute geometry. He practically 

 confines himself to rectilineal figures, and reciprocates 

 segments into angles, thus introducing metrical 

 elements, and becoming necessarily faced by the 

 complications which they involve. It is now familiar 

 to pure mathematicians that, with an " absolute " 

 conic to define our metrical system, there is a con- 

 sistent and reciprocal definition of angle and 

 segment (or distance of two points) by which each of 

 these is the product of a constant and the logarithm 

 of a cross-ratio. But to identify these with the 

 expressions for angle and segment obtained by 

 elementary methods with rectangular coordinates it 

 is necessary to suppose the absolute conic to 

 degenerate into one which, considered as a point- 

 locus, is the line at infinity counted twice, and con- 

 sidered as an envelope is the pair of circular points 

 at infinity. This complicated character of the absolute 

 is at the base of all the puzzling difficulties which 

 beset such attempts as this of Mr. Wilson's — difficul- 

 ties, it is true, which he often surmounts in an 

 ingenious manner. 



For example, in the appendix he introduces a 

 system of angular coordinates, both for lines and 

 points, and obtains point and line equations for the 

 ellipse. Now the unmistakable drift of his thought 

 is that if point and line can be defined by coordinates 

 which measure segments, then " reciprocally " line 

 and point can be determined by coordinates which 

 measure angles. But his angular coordinates are not 

 really reciprocal to the segmental coordinates, as is 

 clear from the fact that his equation of the ellipse 

 is trigonometrical and not algebraical. It might be 

 interesting to decide whether any simple functions 

 of Mr. Wilson's angular coordinates are the direct 

 reciprocals of the ordinary Cartesian segmental 

 coordinates. 



.\ remark should also be made on the note (pp. 

 120-6) on the most general form of the construction 

 of reciprocal figures, as it may prevent possible mis- 

 understanding. In the ordinary process of recipro- 

 cation with an auxiliary conic, F and F' being the 

 corresponding figures, we may say that F' is derived 

 from F by a process, or rule, of polarising, and that 

 F is derived from F' by the same rule. Mr. Wilson 

 gives an example in which F' is derived from F by 

 one process, and F from F' by another — F and F' 

 being reciprocal in the general sense of projective 

 geometry. He adds that this is " wider than the 

 usual method," which, of course, it is, if " the usual 

 method " means employing an auxiliary conic. But 

 the figures obtained by his method can be constructed 

 each from the other by the general method of making 

 four points (or lines) in F correspond at pleasure to 

 four lines (or points) in F', and then to every linear 

 way of constructing F' from F there is a dualistically 

 corresponding way of deriving F from F'. So that 

 it must not be supposed that Mr. Wilson has dis- 

 covered any essentially new way of constructing 

 reciprocal figures, though his remark might be mis- 

 understood in that sense. 



NO. 1877, VOL. 72] 



To return to the more popular aspect of this interest- 

 ing book. The figures are, strictly, strips of black 

 on a white ground. For the author they represent 

 geometrical lines, and are reasoned upon as such. 

 But the reader may give them different interpret- 

 ations, and make up problems for himself accordingly. 

 For example, let the lines in a diagram represent 

 cuts made in a single piece of wood by a fret-saw ; 

 how many pieces are produced? What is the simplest 

 wire model that will give a shadow like a given 

 diagram? and so on. Stencilling, again, is full of 

 problems analogous to those which Mr. Wilson dis- 

 cusses ; knitting and netting give any number of 

 examples of single-path figures. The proverb that 

 " extremes meet " is curiously illustrated by these 

 purely topographical questions, which suggest puzzles 

 for children, problems for designers, and tools for 

 logicians ; while they appear with startling un- 

 expectedness in the most abstruse mathematical 

 theories — Abelian functions, group-theory, hydro- 

 dynamics, and electricity. G. B. M. 



ORGANIC PREPARATIONS AND THE COAL- 

 TAR COLOUR INDUSTRY. 



The Synthetic Dyestuffs and the Intermediate Pro- 

 ducts frotn which they are derived. By J. C. Cain 

 and J. F. Thorpe. Pp. xiv + 405. (London: Chas. 

 Grifiin and Co., Ltd., 11105.) Price i6,t. net. 



THE publication of this work is not without 

 significance in its bearing on the oft-repeated 

 statement that the great industry represented by the 

 manufacture of coal-tar dyes is decaying almost to 

 vanishing point in this country. The fact of publi- 

 cation presumes a demand which, in this case, must 

 be mainly confined to those connected with, or train- 

 ing for, the manufacture referred to. It is unlikely 

 that any great number of students in the colleges of 

 this country are preparing for positions in colour 

 works abroad, and it is therefore reasonable to assume 

 that those concerned with the production of the book 

 have satisfied themselves that the industry is not in 

 such a parlous state as pessimists would have us 

 believe. In any event, the book will pow.erfully 

 influence one factor in the case— the proper instruc- 

 tion of students who are training for the industry. 



Whether this touches the root of the matter is, 

 however, doubtful. The gradual decline in import- 

 ance of the manufacture of coal-tar products in this 

 country has been variously ascribed to the deficient 

 training given in the colleges, the bad patent laws, 

 and the cost of alcohol, relatively to the conditions 

 existing in Germany with regard to these matters. 

 Concerning the work of the colleges, it is now 

 generally conceded that the best of our schools of 

 organic chemistry need fear no comparison with those 

 abroad. The effect of our patent laws, both past and 

 present, in handicapping the industry, has doubtless 

 been 'very great; but possibly the inquiry of a Royal 

 Commission, such as recently reported into the ques- 

 tion of industrial alcohol, would show that, as has 

 been conclusively proved with regard to the cost of 

 alcohol, the effect of the patent laws on the non- 



