Nov. 18, 1881.] 



♦ KNOWLEDGE 



61 



spring. To put the matter more definitely. If the earth rotatoil 

 111 an upi-ight axis, the sun would everywhere, all the year round, 

 rise exactly in the east (apart from the slight effects of atmos- 

 pheric refi-action) and set exactly in the west, attaining at midday 

 an elevation equal to the complement of the latitude— 90° at the 

 I'liuator (which would bring the sun overhead, 70° in latitude 20; 

 111° in latitude 50j 20° in latitude 70 ; and having no elevation at 

 !l in latitude 90, or at cither pole. — Stcdext. 



.S] — Volume of Sphere.— The simplest way of showing that the 



'^'ume of a sphere is two-thirds that of the enclosing cylinder, is 



-ivcn in the following sketch, which, if " Archimedean " is anything 



• a geometrician, he will have no difficulty in tilling in : — Show, 



r<t, that the surface of the sphere is equal to the curved surface of 



" enclosing cylinder, by taking any two planes close to each other 



rough sphere and cylinder, parallel to the top and bottom of 



'lic latter, and showing that the part of spherical surface between 



ihcso planes is equal to the part of cylindi'ical surface between 



t licm. This having been shown, imagine the surface of sphere, thus 



kiiDwn, divided into indetinitely small areas, each of which may be 



> ■ '^'arded as the base of a pyramid, ha'ving the centre of the sphere 



, vertex — all these pyramids together giving the volume of sphere. 



I iiiir combined volume is equal to that of a p\Tamid having a base 



t qual in surface to the surface of the sphere (that is, of its enclosing 



cylinder, by what has been already shown) , and a height equal to 



ilie radius of the sphere. This volume, by well-known property of 



pyramid, is represented by one-third the product of numbers reprc- 



siMiting the curved surface of cylinder and the radius of sphere. 



Hut curved surface of cylinder 



= circumference of base x 2 radius of sphere 

 = circumference of a great circle x 2 radius of sphere. 

 lliuce volume of sphere 



= ^ radius x circumference of a great circle x 2 radius 



2 4-)-'' 



= - (radius)- x 2ir (radius) = — — (if radius = r). 



Hut volume of enolosing cylinder = area of base x 2 height 



= 7rr- x2i- = 2-r». 

 Itcnce volume of sphere is two-thirds that of cylinder. 



Matiie.matuus. 



Effects of IjiGnixixc ox Tkf.es ne.ie a Telegraph Wire. — 

 Some instructive facts in this connection have been brought to 

 light by M. Montigny, in recent examination of poplars bordering 

 part of a road in Belgium, between Eochefort and Dinant. The 

 part in question is some 4,600 metres in length, and runs westward ; 

 it is level for some distance, then rises gradually to a height of 61 

 metres, through a wood, traverses a wooded plateau 200 metres 

 in extent, then descends, still through wood, to a plain. A tele- 

 graph wire runs near the row of Virginia poplars on the north side, 

 and it appears that, out of nearly 500 poplars fcrming this row, 81, 

 or a sixth, have been struck by lightning. Hardly any have been 

 struck in the other row. The trunks have been mostly struck on 

 their south side, and nearly opposite the wire. Comparing different 

 portions of the road, it is found that in the horizontal part none of 

 the (129) trees show injury from lightning, or at most only one (a 

 doubtful case), but as the road rises through the wood, the cases 

 quickly multiply, and on the wooded plateau as many as nine out 

 of 14 trees, or 64 per cent., have been struck. On the slopes the 

 proportion is 25 per cent. ^I. Montigny distinguishes three kinds 

 of injuries — (1) the bark torn and detached on a limited part 

 of the trunk ; (2) a furrow, straight or (rarely) spiral, made 

 on the tree, from near the wire, down to the ground ; and 

 (3) a peculiar oval wound, with longer axis vertical, and lips 

 coloured light brown. Now, the furrows, which are probably due 

 to the most violent discharges, are reUtively most frequent on the 

 platean and on the western slope, which the storms usually reach 

 first. M. Montigny is of opinion that the lightning, while attracted 

 by the wire, does not strike this first, then the tree, but strikes the 

 tree directly. His conception of the process is to the following 

 effect : — Suppose a thunder-cloud charged with positive electricity. 

 A long telegraph wire under it, though insulated, may acquire as 

 great negative tension in the nearest part as if in direct communi- 

 cation with the gi-ound, and the tension is greater the nearer to the 

 cloud. While the inductive influence affects the wire most, near 

 objects, such as trees, share in the influence according to their con- 

 ducting power. The lightning, attracted in the direction of the 

 wire, yet does not strike this, the insulating cups presenting an 

 obstacle to its prompt and rapid escape. It finds a better conductor 

 to earth in a neighbouring poplar, wet with rain. From the facts 

 indicated it results, that of two similar houses, one built on a plain, 

 the other in <a wood, .and having a telegraph wire fixed to them, the 

 latter is the more liable to injury by lightjiing, and the danger is 

 greater if the wood enclosing the house be upon an eminence. — 

 rimes. 



(Buv iWatftrmatiral Column. 



FROM the way in which logarithms are commonly spoken of, one 

 would suppose that they were originally intended to perplex 

 the student, instead of having been devised specially to assist him. 

 It may be that this is due chiefly to the use of a name whose real 

 moaning is not known, while, were its real meaning known, the uso 

 of numbers so named would still remain a mysterj- to all save 

 mathematicians. The word logarithm is really intended to signify 

 ratio-number. But hundreds who uso logarithms, and thousands 

 who would do well to use them, would not be one whit the wiser for 

 knowing that there are tables of numbers which may be regarded 

 as ratio-numbers. If a name had been given to logarithms which 

 suggested to all something of their real use to computers, we should 

 find tliem more valued and more commonly employed than they 

 arc. But unfortunately the singular idea that nothing but a long 

 unintelligible name is worth anything in science — an idea about as 

 worthy of respect as the liarbadocs mother's admiration for the 

 name Chrononhotontliologos which Captain Slarryat gave to her 

 baby — has caused these useful tables to bear a ridiculous because 

 un-English name. 



In reality, a logarithm is a ratio number, though that says very 

 little to most men of its use. The logarithms of the numbers 1, 2, 

 3, 4, 5, &c., up to 100,000 in our books, are in reality the powers to 

 which 10 must be raised to give the numbers 1, 2, 3, 4, 5, &c., 

 respectively. If we know the power to which 10 must be raised to 

 give the number 13, and also the power to which 10 must be raised 

 to give the number 17, we have only to add these powers together 

 to obtain the power to which 10 must bo raised to give the product 

 of 13 by 17, or 221. If our tables give all the powers of numbers 

 from 1 to 221, wo need not actually multiply together 13 and 17 to 

 get their product 221. All we need do is to add the logarithm of 

 13 to the logarithm of 17 ; the sum is the logarithm of 13 and 17 

 multii)lied together. We look out in our tables the logarithm cor- 

 responding to the sum of the logarithms of 13 and 17, and we find 

 that the number corresponding to this logarithm is 221. 



Here, of course, wo have not been saved a particle of labour. 

 While we were looking out these logariihms, a charity boy could 

 have mnltiplicd 13 and 17 together half a-dozen times. 



But, supposing we had occasion to nuiltiplv together the num- 

 bers 21,714, and 36,912, and to divide the product by 78,124 and 

 02,315 ; that is, to divide the product of the first pair of numbers 

 by the product of the second pair of numbers. Then, if we know 

 the powers to which 10 must be raised to give the above four 

 numbers respectively, we can tell, by simple addition and subtrac- 

 tion, the power to which 10 must be raised to give the answer to our 

 little sum ; and if we know also what is the number which would 

 result from raising 10 to the power just mentioned, this ntimber is 

 the answer we require. Suppose, for example, that 10» = 21,714; 

 10" = 56,912; 10'^ = 78,124; 10" = 62,315. Then we know by the 

 properties of ratios that 



21714 X. 56012 = 10""'"'' 



7812-1x62315 = 10 



H rf 



while 



21714 X 56912 

 78124 X 62315 



= 10" 



Now a table of logarithms gives us for all numbers from 1 to 

 100,000 the powers corresponding to a, 5, c, and d in the above 

 example. (We can find such powers also easily enough from the 

 tables for all numbers from 100,000 up to 10,000,000.) So that aU 

 the computer in the above case would have to do would be to look 

 out the powers con-csponding to a, h, c, and d, to add a to 6 and 

 c to (J, subtracting then the latter sum from the former. The result 

 would be the power to which ten must be raised to give the answer 

 to his sum ; and he would only require to find that power in his 

 tables to get the number he wanted. Ue would, probably, do all 

 this in about the time that a first-rate computer would have got 

 half way through the multiplying of 21,714 and 56,912 together. 



This is, of course, only a general account of the use of a book of 

 such powers, or of a book of logarithms. It should suflice to shcT 

 that, despite their hard name, they arc worth understanding by all 

 who have much computing to do ; and there are few who would not 

 do well, by studying a little the use of logaritluns, to make them- 

 selves ready to em[iIoy the tables when occasion may arise. 



Turn now to a few details. 



The logarithms given in the tables aro in reality the logarithms of 

 the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 ; 1, 11, 1-2, 13, 1-4, 15, IG, 

 17, 18, 19; 2, 2'1, &c. ; that is, however large the number in the 

 letthand column, it is understood to lie between 1 and 10, so that 

 its logarithm lies in value between and 1 (ten to the power nought 

 gives 1, and ten to the power 1 is ten). Decimal points are not 



