Aug. 1, 1884] 



♦ KNOWLEDGE 



101 



BO divisible. If there be a remainder, it will either be the true 

 remainder or its oompleraent, according as the sum of the plus 

 terms is greater or less than that of the minus terms. 



The extreme left-hand term may consist of less than three 

 figures. William Si.vgers. 



LETTERS RECEIVED AND SHORT ANSWERS. 

 J. Greevz Fisiiek. Forgive me. I do not "discountenance 

 improvements in spelling because reformers are not unanimous," 

 but because I regard the ortliography in which is enshrined the 

 noble diction of Shakespeare, Milton, Pope, Addison, Macaulay, 

 and Tennyson as quite good enough for me. As to the destruction 

 of all traces of the etymology of words by " fonctic vagariz," I 

 need say nothing to any philologist. I am much more disposed to 

 sympathise with you in your protest against the meaningless limi- 

 tations of the halfpenny postage. — Hector. The so-called " storm- 

 glasses" are simply glass tubes (not air-tight) filled with a solution 

 of camphor, nitrate of potash, and sal ammoniac. They are hygro- 

 meters, if they are anything, but both light and temperature affect 

 them. For any scientific purpose they are worthless. — Wm. Wilsox, 

 M.A., LL.D. Accepting Colenso's example as one illustrating 

 Common Measure, yon are obviously right in what you say as to his 

 definition. — D. Wixstaxley dreamt on July Sth that his ex-sweet- 

 heart came to say " good-bye" on the very morning on which (as 

 he subsequently discovered) she was married to some one else. He 

 also dreamt, on the 1-tth, a medical man but slightly known 

 to him died on the floor of his (Mr. W.'s) bedroom. This gentle- 

 man did die, in his own bed, early the next morning. My 

 correspondent goes on to suggest that certain sorts of suppers 

 may " yield prophetic dreams," which is too much for me. — 

 Edward F. Hoerale. Without yourself taking the trouble to 

 attend any course of lectures, you will, I think, find in Weinhold's 

 " Introduction of Experimental Physics " ample material for such 

 instruction as you propose to impart ; and you might even take the 

 successive chapters of " Facts Aronnd Us " — a little book published 

 by Stanford's — as the basisjof a series of lectures to them. I cannot 

 possibly make an appointment for an interview, nor undertake to 

 see any one personally in connection with matters discussed in these 

 columns. — Thomas Maclean. Does it ever strike you that I have 

 scores — not to say hundreds — of correspondents to deal with, and 

 that in the tremendous pressure which exists upon an editor's 

 time, it is quite possible (and even excusable) for him to overlook 

 papers among those which descend in shoals on his devoted head ? 

 I have not communicated your notions to Messrs. Browning and 

 Wenham. You must do so yourself if you wish them brought 

 before those gentlemen. — William Worslet. I do not print the 

 letter on cholera which you send, inasmuch as it contains internal 

 evidence that the " Rev. Dr." who penned it is ignorant of the very 

 rudiments of physiology and pathology. — Ax Exile, while pro- 

 testing against the selection of Ireland as an illustration of 

 a country sunk in the lowest depths of superstition, points out, 

 not wholly without reason, that past misgovernment may be 

 to some extent held to be at the bottom of such a deplorable 

 state of things. — Oxford asks this "question": Is it possible to 

 obtain 360 different positions of the letters a, b, c, d, e, f, by means 



of the following two methods of transposition only, (i) 



and (ii) 



a, h, c, d, e, f 



a, b, c, d, e, f 

 a, d, b, f, c, e 



, the last position being a, i>, c, d, e, f ? We 



/, d, a, e, b, c 



commend this to readers with plenty of idle time on their hands. — 

 Edward .S. Haxsox points out that the " Design for a Parlour 

 Organ," which we extracted from the Scientific American on p. 

 5S, was copied from the Building Neios into the EngHsh Mechanic 

 of March 28 last, and was appropriated by the American paper 

 without any acknowledgment at all. Thanks for your details anent 

 the English translation of Fontenelle. — E. D. Warring. When a 

 man submits his views for publication, he, ipso factor invites 

 criticism upon them. You surely wanted my candid opinion of 

 yours, and not an utterly insincere compliment. — Fred. Jackmax. 

 WiU you kindly read the paragraph at the bottom of the first 

 column on p. 62, where if is definitely stated that I have given 

 up lecturing altogether ? — Kemts. I am unfortunately unable to 

 give the address of the Cremation Society. Perhaps some reader 

 of Knowledge will kindly do so. — J. H. Hayward. I am, I regret 

 to say, out of town. — St. E. You can scarcely do better than get 

 the " Human Physiology," by Dr. W. B. Carpenter. — Eye- Wit- 

 ness. I do not think that either of my works which you name 

 would render you more assistance than those which you possess. 

 "The Stars in their Seasons " is a reprint of the maps which 

 appeared monthly in Vols. I. and II. of Kxowledge. If you merely 

 want the old constellation figures, get the " Six Star Maps on the 

 Gnomonic Projection," now published by Letts's, but originally 

 issued many years ago by S.D.U.K. 



0av iHatljematiral Columm 



EASY LESSONS IX CO-ORDINATE GEOMETRY. 



By Richard A. Proctor. 



Problems ox the Straight Line. 



"TTTE proceed to discuss the equations to lines fulfilling given con- 

 V 1 ditions, and also to examine some problems with which it is 

 necessary the student should be familiar. 



61. Prop. — To determine the form of the equation to a straight line 

 passing through a ijiven point. 



Let x' y' be the co-ordinates of the given point. Now we have 

 already seen that the equation to a straight line passing through 

 x' y', and inclined at an angle whose tangent is m to the axis of 

 X is 



y — y^ = m (x— i') 



(0 



We might have obtained this equation as follows : — Let the equation 

 to the required line be 



y = mx + c (ii) 



then, since the line passes through the point x' y', equation (i) must 

 be satisfied when x ' is written for x, and y* for y ; therefore 



y' = mx'-^c (iii) 



And subtracting (iii) from (ii) we clearly obtain equation (i). This 

 amounts to the elimination of c between equations (ii) and (iii). 

 Instead of eliminating c we might have eliminated m ; thus, sub- 

 stituting for m in (ii) the value given by equation (iii) we get 



v' — c 



this represents a straight lino passing through the point x' y' and 

 having an intercept c on the axis of x. 



The equation will clearly assume different forms according to the 

 condition we suppose the straight line to fulfil besides passing 

 through the given point. We shall now consider the equations of 

 lines fulfilUng such conditions. It is well to note in passing, how- 

 ever, that whatever form of equation to the straight line be 

 adopted in place of equation (ii), we obtain when the constant 

 term is eliminated by subtraction, an equation of the form 

 x-x' + l{y-y') = 



Thus if in place of (ii) we take the equation 



y , 



we get in place of (iii) 



or subtracting 



a b 



a b 



• X' y • 



= 



similarly from the equation 



X cos a -i- y sin a = p 

 we should obtain 



(x — x') cos a + {y — y') sin a = 

 and finally from the equation 



Ax-hBy-hC = 

 we should obtain 



A (.r-x')-HB(!/-v')=0. 



It is important that the student should note these results. They 

 show that in this particular case a relation subsists which we shall 

 presently show to hold generally. The equations 



X— a?' =0, and y — y'=0 

 represent lines whose intersection determines the point x' y', and 

 it appears that the equation formed by combi nin g these two 

 equations into a single equation of the forms 



X — x' -I- a const, (y — 'y')=0, 

 represents a straight lino through the intersection of the lines 

 represented by the two equations so combined. 



62. Prop. — To determine the equation to a straight line passing 

 throxigh two points. 



Let Xi !/i be the co-ordinates of one of the given points, X; y, 

 those of the other. Then it follows from the preceding article that 

 the equation of a straight line through xi yi is of the form 



V — y\ = m (r — .T,) (i) 



where m is the tangent of the angle which the straight line makes 

 with the axis of x. If the straight line represented by (i) passes 



