JoNK 30, 1882. 



• KNOWLEDGE 



the eyelenB Oil inch. These lenses must be placed 23 inch 

 apart with, we need scarcely add, their plane sides towards the eye 

 of the observer.— Lancasiuee will find that selenography will supply 

 him with work at once interesting and useful. The study of the 

 Solar surface, too, may be pursued with an instrument of the size 

 of his, and the addition to it of a Browning's Star spectroscope 

 will enable him to see something of the spectra of the prominences 

 on the solar limb, to say nothing of its use in Stellar spectroscopy. 

 Should ho possess artistic taste, and be able to sketch, careful 

 drawings of such planets as Mars, Jupiter, and Saturn, if made 

 at intervals, possess a direct scientific value. His best plan, 

 however, wi'l bo to purchase forthwith Webb's admirable 

 " Celestial Objects for Common Telescopes," which is in 

 reality a pocket Encyclopaedia for the amateur observer. — 

 KxCEtsiOB. Ilerculis, as viewed on the meridian, with an ordinary 

 inverting eye-piece, now presents the appearance of a large star, 

 ^vith a very much smaller companion almost horizontally to the left 

 of it. We have not a copy of " Half Hours " at hand ; but cer- 

 tainly no such change as your diagrams indicate has taken place in 

 this star. Its position angle has varied some 22° during the last 

 eighty-two years. Prior to the year 1871, the comes was below a 

 horizontal line to the left of its primary ; now it is just above such 

 a line. We do not know upon what authority Castor is described 

 as a ternary system in Nollwyn's " Astronomy." At any rate, no 

 sensible change has taken place in the relative position or distance 

 of the principal star and the 11' magnitude one since the year 1823; 

 and the system is universally regarded among astronomers as being 

 simply binary, the minute star having obviously no physical con- 

 nection with" the well-known pair. With reference to Pollux, the 

 word "coarse" was used as expressive of th*^ < — .,^^,...^l.^.. .i.'^tnnce 

 of the comites ; "fine," as referring to the i ^^■hen 



Pollux is on the meridian on a dark night, yi t two 



companions up, both to the right, and a lii:,' i i ' :;.iiiital 



line passing through their primary. The larger slur it tiie two is 

 about half as far again from Pollux as the smaller one. This last 

 will require some looking for ; though, when caught, it may be held 

 steadily enough in a good three-inch telescope. The earliest recorded 

 position of comet Wells is on March 19 6894, 1882, G.M.T. ; when 

 it was situated in K.A. 17° 54' 38-1", and declination 33° 25' 5" 

 X. (which possibly ought to be 33° 24' 5" N.). 



Long Distance Teiephoning. — Recent experiments have been 

 made with M. van Bysselberghe's system of telephony, during which 

 messsages were sent by telephone over a long distance simultane- 

 ously with telegraphic messages traversing the same wire. The 

 first experiment was made on an ordinary telegraph wire 353 kilo- 

 rafttres long, between Paris and Nancy ; and the second between 

 Paris and Brussels, over a wire 344 kilometres long. It is stated 

 that telephonic messages were distinctly heard, whilst the tele- 

 graphic messages were distinctly rendered by the Morse instru- 

 ments employed. M. van Eyssclberghe is the head of the Meteoro- 

 logical Observatory at Brussels. 



Electbic Light in the City.— At a meeting of the Commissioners 

 <if Sewers, an abstract of the engineers' report was submitted, from 

 which it appears that the cost for twelve months, including the 

 filing of machinery, lamps, Ac., ranges, in district No. 1, from 

 €2,190 to £'5,750, as against the present cost of gas, £551 ; district 

 No. 2, from £2,350 to £4,270, as against £363 for gas ; district No. 

 3, from £2,470 to £3,800, as against £341 for gas ; district No. 4, 

 from £2,920 to £4,350, as against £612 for gas. Some of the com- 

 panies couple with their tender a request to be allowed to light 

 private premises within the district, and in that case the company 

 offer to make the charge for lighting, irrespective of the machinery, 

 4c., the same as for the gas saved plus 20 per cent. These figures 

 show a great disparity when conjparcd with other contracts. 



The Electric Light at Che-sterpielp. — The bill for the public 

 lighting of the streets of Chesterfield since November 1 last, when 

 Messrs. Hammond & Co. undertook to illuminate the thoroughfares 

 of the town by means of electricity, has been received by the 

 Corporation. The period it covers is nearly five months, and the 

 total amount charged £272. During a great part of that time 

 there have been twenty-two arc lights and from si.xty to seventy 

 incandescent lamps in use, as well as gas-oil lamps. On the whole 

 the town has been very fairly lit throughout that period, and at 

 present the light supplied is excellent. The amount for public 

 lighting ]ier annum under the old system was £920. The work was 

 necessitated by the cutting off of the gas supply by the local 

 authorities, thereby leaving the town in darkness. Mr. Kingslnnd, 

 who was the superintending electrician, has boon appointed 

 electrical engineer to the Yorkshire Brush Electric Light 

 Company. 



#ur i¥latt)ematiral Column. 



TO OCR READERS. 



I HAVE been in some doubt whether to take first the series of 

 papers on the solution of Geometrical Problems, or those which 

 I promised in an early number, on the Differential Calculus. After 

 some consideration, however, I have decided to take the latter first, 

 because lessons on the solution of Geometrical Problems can be 

 obtained more easily than instruction really elementary and simple 

 in the Differential and Integral Calculus. In the " Easy Lessons " 

 now commenced, 1 propose to depart in some degree from a plan 

 which I followed in dealing with the Differential Calculus in the 

 pages of the English Mechanic some ten or twelve years ago. In 

 pursuance of a plan suggested to me by the ReT. Mr. Griffin, of 

 Ospringe, 1 propose to bring the Integral Calculus before the 

 reader at the same time as tho Differential Calculus, instead of 

 dealing with it later on. The connection between the two subject.i 

 will thus be more clearly recognised. I propose also to illustratt- 

 geometrically as many as possible of the relations involved in dealing 

 with the Differential and Integral Calculus, believing that in this 

 way the real meaning and value of the methods employed will be 

 more clearly recognised, while the interest of the subject will bo 

 enhanced. 



I may notice in passing that space and time can no longer be 

 devoted to problems set for solution by our readers — that is to say, 

 to problems thus set in the way of puzzles. So many solutions, 

 good, bad, and indifferent, are sent for each such problem, that the 

 work of analysing and comparing them so as finally to publish the 

 best, while awarding to each its due meed of praise, is more than, 

 with our present staff, we are prepared to accomplish satisfactorily. 

 We shall always be pleased, however, to solve, or to publish for solu- 

 tion, problems which may have presented difficulties to students of 

 various mathematical subjects, provided always that such problems 

 have real interest and value, and aronot merely difficult. 



EASY LESSONS IN THE DIFFERENTIAL CALCULUS. 

 No. I. 



Tlie differential calculus is the science which deals with the rate 

 at which variable quantities increase or diminish. When we say 

 that a quantity is variable, we imply that it varies as some other 

 quantity changes. For example, tho velocity of a train is variable. 

 It varies with the iime which has elapsed since tho train started — 

 it varies with the distance traverstd — with the s/eam jx>u**r em- 

 ployed— with the slate o/ the rails — and so on. But the differen- 

 tial calculus deals only with those quantities which vary according 

 to some definite law. 



For example, when a body is let fall from rest tho distance it 

 traverses varies, according to a known law, with tho time elapsed 

 since the fall began. The differential calculus is able to deal 

 with such a case as this. Again the sine of an angle varies accord- 

 ing to a known law as the angle changes ; and the differential cal- 

 culus is therefore able to deal with this case also. 



Now we can at once see the importance of a calculus which will 

 deal with variable quantities. Algebra and geometry and trigo- 

 nometry deal with absolute quantities. But it is often verj- 

 necessary to learn something about the variations of quantities, to 

 know when a variable quantity attains its greatest value, when it 

 is increasing, when diminishing, when it changes fastest, and so 

 on. Whenever variations take place according to a known law, 

 this is precisely what the differential calculus will do for us. And 

 its great advantage is that it will solve our problems systematically. 

 An ingenious application of algebra or geometry or trigonometry 

 will often enable us to solve problems which belong especially to 

 the differential calculus. But we require ingenuity for the purpose, 

 whereas the differential calculus solves such problems with cer- 

 tainty, even if wo have not a particle of ingenuity, so long only as 

 we follow tho proper rules. Even where it fails, it teaches us that 

 we are trying to solve an insoluble problem. 



The first matter the calculus attends to is tho choice of a con- 

 venient expression for the rate at which a rarioblo quantity 

 charges. This expression is called a differential coefficinrt. 1 prefer 

 to illustrate rather than to define it. I wish also to illustrate it in 

 such a way as to remove at the outset the chief stumbling block of 

 the student of this special department of mathematics. 1 take, 

 therefore, a familiar case of a varying quantity : — 



When a body is let fall from rest, wo know that as it falls its 

 velocitv continually increases. Now this varying velocity affords • 

 very good illustration of a differential coefficient. The velocity of 

 a body may bo described as the rate with which the 8|>aco it ho» 

 traversed is increasing, as tho time elapsed increases. When we 

 change the time, wo change the space traversed. But unless the 

 velocity is uniform, tho change of space is not proportional to the 

 change of time. In the case of a falling body, tho velocity is not 



