I. 32 and Euclid I. 47 can be so supplemented and are valid.

On the other hand, as an illustration of how deceptive a non- mathematical proof may be, I here mention the familiar paradox that a square of paper, subdivided like a chessboard into 64 small squares, can be cut into four pieces which being put
CH. Ill]
GEO:\IETUICAL RECREATIOMS
63
together form a figure containing 65 suoh small squares*. This is effected by cutting the original square into four pieces in the manner indicated by the thick lines in the first figure. If these four pieces are put together in the shape of a rectangle in the way shown in the second figure it will appear as if this rectangle contains 65 of the small squares.
This phenomenon, which in my experience non-mathema- ticians find perplexing, is due to the fact that the edges of the four pieces of paper, which in the second figure lie along
__J-
"7 ^
t-ZZ
===/
•^
■^
"^
^^
■*>,
\
■^^^
•^
^
^
B
the diagonal AB, do not coincide exactly in direction. In reality they include a small lozenge or diamond-shaped figure, whose area is equal to that of one of the 64 small squares in the original square, but whose length AB'\^ much greater than its breadth. The diagrams show that the angle between the two sides of this lozenge which meet at A is tan~^| — tan~^|, that is, is tan~^^, which is less than 1J°. To enable the eye to distinguish so small an angle as this the dividing lines in the first figure would have to be cut with extreme accuracy and the pieces placed together with great care.
This paradox depends upon the relation 5x13 — 8^=1. Similar results can be obtained from the formulae
13x34-212 = 1, 34x89-552=1,...;
or from the formulae
5--3x8 = l, 132-8 X 21 = 1, 342-21 x55 = l,....
* I do not know who discovered this paradox. It \i given in various modem books, but I cannot find an earlier reference to it than one in the Zeitachrijt fUr Mathematik und Fhysik, Leipzig, 1868, vol. xiii, p. 102.
54 GEOMETRICAL RECREATIONS [CH. Ill
These numbers are obtained by finding convergents to the con- tinued fraction
111 1+ 1 4.1 + 1 + —•
A similar paradox for a square of 17 cells, by which it was shown that 289 was equal to 288, was alluded to by Ozanam*, who gave also the diagram for dividing a rectangle of ll by 3 into two rectangles whose dimensions appear to be 5 by 4 and 7 by 2.
Turtons Seventy - Seven Puzzle. A far better dissection puzzle was invented by Captain Turton. In this a piece of cardboard, 11 inches by 7 inches, subdivided into 77 small equal squares, each 1 inch by 1 inch, can be cut up and re-arranged so as to give 78 such equal squares, each 1 inch by 1 inch, of which 77 are arranged in a rectangle of the same dimensions as the original rectangle from one side of which projects a small additional square. The construction is in- genious, but cannot be described without the use of a model. The trick consists in utilizing the fact that cardboard has a sensible thickness. Hence the edges of the cuts can be bevelled, but in the model the bevelling is so slight as to be imperceptible save on a very close scrutiny. The play thus given in fitting the pieces together permits the apparent pro- duction of an additional square.
Colouring Maps. I proceed next to mention the geo- metrical proposition that not more than four colours are neces- sary in order to colour a map of a country {divided into districts) in such a way that no two contiguous districts shall he of the same colour. By contiguous districts are meant districts having a common line as part of their boundaries: districts which touch only at points are not contiguous in this sense.
The problem was mentioned by A. F. Mobiusf in his Lectures in 1840, but it was not until Francis Guthrie J com-
* Ozanam, 1803 edition, vol. i, p. 299.
t Leipzig Transactions {Math.-jyhys. Glasse), 1885, vol. xxxvir, pp. 1—6. :J: See Proceedings 0/ the Royal Society of Edinburgh, July 19, 1880, vol. x, p. 728.
CH. Ill]
GE(JMETRICAL REOIIEATIOXS
municated it to De Morgan about 1850 that attention was generally called to it : it is said that the fact had been familiar to practical map-makers for a long time previously. Through De Morgan the proposition then became generally known ; and in 1878 Cayley* recalled attention to it by stating that he could not obtain any rigorous proof of it.
Probably the following argument, though not a formal demonstration, will satisfy the reader that the result is true. Let A, B, G be three contiguous districts, and let X be any other district contiguous with all of them. Then X must lie either wholly outside the external boundary of the area
ABC or wholly inside the internal boundary, that is, it must occupy a position either like X or like X\ In either case there is no possible way of drawing another area Y whicli shall be contiguous with A,B,G, and X. In other words, it is possible to draw on a plane four areas which are contiguous, but it is not possible to draw five such areas. If ^, By C are not con- tiguous, each with the other, or if X is not contiguous with A, B, and G, it is not necessary to colour them all differently, and thus the most unfavourable case is that already treated.
* Proceedings of the London Mathematical Society, 1878, vol. ix, p. 148, and Proceedings of the Royal Geographical Society, London, 1879, N.S., vol. i, pp. 259-261, where some of the difificulties ar« indicated.
56 GEOMETRICAL RECilEATlONS [CH. Ill
Moreover any of the above areas may diminish to a point and finally disappear without affecting the argument.
That we may require at least four colours is obvious from the above diagram, since in that case the areas A, B, G, and X would have to be coloured differently.
A proof of the proposition involves difficulties of a high order, which as yet have baffled all attempts to surmount them. This is partly due to the fact that if, using only four colours, we build up our map, district by district, and assign definite colours to the districts as we insert them, we can always contrive the addition of two or three fresh districts which cannot be coloured differently from those next to them, and which accordingly upset our scheme of colouring. But by starting afresh, it would seem that we can always re-arrange the colours so as to allow of the addition of such extra districts.
The argument by which the truth of the proposition was formerly supposed to be demonstrated was given by A.B.Kempe* in 1879, but there is a flaw in it.
In 1880, Tait published a solution f depending on the theorem that if a closed network of lines joining an even number of points is such that three and only three lines meet at each point then three colours are sufficient to colour the lines in such a way that no two lines meeting at a point are of the same colour; a closed network being supposed to exclude the case w^iere the lines can be divided into two groups between which there is but one connecting line.
This theorem may be true, if we understand it with the limitation that the network is in one plane and that no line
* He sent his first demonstration across the Atlantic to the American Journal of Mathematics, 1879, vol. ii, pp. 193 — 200; but subsequently he communicated it in simplified forms to the London Mathematical Society, Transactions, 1879, vol. X, pp. 229—231, and to Nature, Feb. 26, 1880, vol. xxi, pp. 399—400. The flaw in the argument was indicated in articles by P. J. Heawood in the Quarterly Journal of Mathematics, Loudon, 1890, vol. xxiv, pp. 332 — 338; and 1897, vol, XXXI, pp. 270—285.
t Proceedings of the Royal Society of Edinburgh, July 19, 1880, vol. x, p. 729 ; Philosophical Magazine, January, 1884, series 5, vol. xvu, p. 41 ; and Collected Scientific Papers, Cambridge, vol. ii, 1800, p. 93.
CH. Ill] GEOMETUICAL RECUEATIONS 57
meets any other line except at one of the vertices, \vhich is all that we require for the map theorem; but it has not been proved. Without this limitation it is not correct. For instance the accompanying figure, representing a closed network in three dimensions of 15 lines formed by the sides of two pentagons and the lines joining their corresponding angular points, cannot be coloured as described by Tait. If the figure is in three dimensions, the lines intersect only at the ten vertices of the network. If it is regarded as being in two dimensions, only the ten angular points of the pentagons are treated as vertices of the network, and any other point of intersection of
the lines is not regarded as such a vertex. Expressed in tech- nical language the difficulty is this. Petersen* has shown that a graph (or network) of the 2nth. order and third degree and without offshoots (or feuilles) can be resolved into three graphs of the 2?ith order and each of the first degree, or into two graphs of the 2nth. order one being of the first degree and one of the second degree. Tait assumed that the former resolution was the only one possible. The question is whether the limitations mentioned above exclude the second resolution.
Assuming that the theorem as thus limited can be estab- lished, Tait's argument that four colours will suffice for a map is divided into two parts and is as follows.
* See J. Petersen of Copenhagen, VIntermedinire des MatMmaticiens, vol. v, 1898, pp. 225—227; and vol. vi, 1899, pp. 30—38. Also Acta Madiematica, Stockholm, vol. xv, 1891, pp. 193—220.
58 GEOMETRICAL RECREATIONS [CH. Ill
First, suppose that the boundary lines of contiguous dis- tricts form a closed network of lines joining an even number of points such that three and only three lines meet at each point. Then if the number of districts is 71 + 1, the number of boundaries will be Sn, and there will be 2n points of junction; also by Tait's theorem, the boundaries can be marked with three colours /3, 7, 8 so that no two like colours meet at a point of junction. Suppose this done. Now take four colours, A,B,G, D, wherewith to colour the map. Paint one district with the colour A; paint the district adjoining A and divided from it by the line y8 with the colour B ; the district adjoining A and divided from it by the line 7 with the colour 0; the district adjoining J. and divided from it by the line B with the colour I). Proceed in this way so that a line /3 always separates the colours A and B, or the colours G and D; a line 7 always separates A and (7, or D and B; and a line S always separates A and D, or B and G. It is easy to see that, if we come to a district bounded by districts already coloured, the rule for crossing each of its boundaries will give the same colour : this also follows from the fact that, if we regard ^, 7, 8 as indicating certain operations, then an operation like B may be represented as equivalent to the effect of the two other operations y8 aad 7 performed in succession in either order. Thus for such a map the problem is solved.
In the second case, suppose that at any point four or more boundaries meet, then at any such point introduce a small district as indicated below: this will reduce the problem to the first case. The small district thus introduced may be
>c
coloured by the previous rule; but after the rest of the map is coloured this district will have served its purpose, it may be then made to contract without limit to a mere point and will disappear leaving the boundaries as they were at first.
Although a proof of the four-colour theorem is still wanting, no one has succeeded in constructing a plane map which requires
CH. Ill] GEOMETRICAL RECREATIONS 59
more than four tints to colour it, and there is no reason to doubt the correctness of the statement that it is not necessary to have more than four colours for any plane map. The number of ways in which such a map can be coloured with four tints has been also considered*, but the results are not suffi- ciently interesting to require mention here.
I believe that in the corresponding question with solids in space of three dimensions not more than six tints are required to colour the exposed surfaces, but I have never seen any attempt to prove this extension of the problem.
Physical Configuration of a Country. As I have been alluding to maps, I may here mention that the theory of the representation of the physical configuration of a country by means of lines drawn on a map was discussed by Cayley and Clerk Maxwellf. They showed that a certain relation exists between the number of hills, dales, passes, &c. which can co-exist on the earth or on an island. I proceed to give a summary of their nomenclature and conclusions.
All places whose heights above the mean sea level are equal are on the same level. The locus of such points on a map is indicated by a contour-line. Roughly speaking, an island is bounded by a contour-line. It is usual to draw the successive contour-lines on a map so that the difference between the heights of any two successive lines is the same, and thus the closer the contour-lines the steeper is the slope, but the heights are measured dynamically by the amount of work to be done to go from one level to the other and not by linear distances.
A contour-line in general will be a closed curve. This curve may enclose a region of elevation: if two such regions
• See A. G. Dixon, Messenger of MatJiematics, Cambridtje, 1902-3, vol. xxxii, pp. 81—83.
t Cayley on ' Contour and Slope Lines,' VhiloRophical Marjazine, London, October, 1859, eeiies 4, vol. xvui, pp. 264 — 268 ; Collected Works, vol. iv, pp. 108 — 111. J. Clerk Maxwell on 'Hills and Dales,' Philosophical Magazine, December, 1870, series 4, vol. xl, p|). 421—127; Collected Works, vol. ii, pp. 233—240.
60 GEOMETRICAL RECREATIONS [CH. Ill
meet at a point, that point will be a crunode (i.e. a real double point) on the contour-line through it, and such a point is called a pass. The contour-line may enclose a region of de- pression: if two such regions meet at a point, that point will be a crunode on the contour-line through it, and such a point is called a fork or bar. As the heights of the corre- sponding level surfaces become greater, the areas of the regions of elevation become smaller, and at last become reduced to points: these points are the summits of the corresponding mountains. Similarly as the level surface sinks the regions of depression contract, and at last are reduced to points: these points are the bottoms, or immits, of the corresponding valleys.
Lines drawn so as to be everywhere at right angles to the contour-lines are called lines of slope. If we go up a line of slope generally we shall reach a summit, and if we go down such a line generally we shall reach a bottom: we may come however in particular cases either to a pass or to a fork. Districts whose lines of slope run to the same summit are hills. Those whose lines of slope run to the same bottom are dales. A watershed is the line of slope from a summit to a pass or a fork, and it separates two dales. A watercourse is the line of slope from a pass or a fork to a bottom, and it separates two hills.
\i n-\-l regions of elevation or of depression meet at a point, the point is a multiple point on the contour-line drawn through it; such a point is called a pass or a fork of the ni\i order, and must be counted as n separate passes (or forks). If one region of depression meets another in several places at once, one of these must be taken as a fork and the rest as passes.
Having now a definite geographical terminology we can apply geometrical propositions to the subject. Let h be the number of hills on the earth (or an island), then there will be also h summits ; let d be the number of dales, then there will be also d bottoms ; let p be the whole number of passes, pi that of single passes, p^ of double passes, and so on ; let / be the whole number of forks, /i that of single forks, f of double
CH. Ill] GEOMETRICAL RECREATIONS (>1
forks, and so on ; let w be the number of watercourses, then there will be also w watersheds. Hence, by the theureuis of Cauchy and Euler,
A = 1 + j9j + 2j[;., + . . . ,
and w=^ (p, +/i) + 3 (p^ +/;) 4- ....
These results can be extended to the case of a multiply- connected closed surface.
ADDENDUM.
N'ote. Page 52. The required rotation of the lamina can be effected thus. Suppose that the result is to be equivalent to turning it through a right angle about a point 0. Describe on the lamina a square OAlJC, Rotate the lamina successively through two right angles about the diagonal OB as axis and through two right angles about the side OA as axis, and the required result will be attained.
K9