Aug. 8, 1884.] 



♦ KNOWLEDGE ♦ 



123 



have the valuable space of Knowledge wasted in the discDSEion of 

 an imposture which is simply a disprace to the boasted intellectual 

 advancement of the nineteeutli century. — Fides. I regret to say 

 that I am ipjnorant of the correspondence class to which yon refer. 

 It was apparently a private adventure, possessing no official cha- 

 racter whatever. — SvnNEV Pockli.nc.tox. To give the objects 

 described as observed with a three-inch telescope, as seen in a two- 

 inch one, would be simply to show them a very little smaller, and 

 not quite so well-defined ; the closest separable double stars, for 

 example, in the large instrument merging into misshapen single 

 ones in the smaller telescope. It would be the veriest waste of 

 space. — Ax AxoxiMors CuKRE.spoxnEXT sends us the Times obituary 

 for July 29, as containing this coincidence : — " On the 25th inst., at 

 Delverton-road, Jlanor-place, Walworth, Jolin Li'jhfning Morgue, 

 aged 75," and " On the 26th July, after three years' suffering, 

 Margaret, wife of Captain George Thunder, aged 31. ' These names 

 are sufficiently uncommon to read oddly in such connection.— A 

 Constant Reader. I have nothing to add to what I said about 

 Professor Loisette's system in Jso. 117. Vou must hence judge of 

 its applicability to your own case. If by " Abstract English 

 History," you mean a cram-book, I can recommend none such. 

 Dr. Wm. Smith's " Smaller English History," though, ought to 

 answer your purpose. Stewart's " Modem Geography," too, pub- 

 lished by Oliver & Boyd, will give you a sound grounning in the 

 subject of which it treats. — H. Francis. Your "Coincidence" is of 

 too doubtful a character to warrant me in wasting space by its 

 insertion. You will see your extracts from the Christian Glohe 

 elsewhere. There is not a trace either of air or water npon the 

 Lunar surface. Did the latter exist, its evaporation must, perforce, 

 form clouds. Of course, the Moon appears inverted as viewed from 

 the Southern hemisphere, i.e., when wo are looking at, say, the 

 "Metropolitan Crater" Tycho, in England with the naked eye, it 

 appears to be at the bottom of the Moon; while, as ■■■iewed from 

 New Zealand, it is at the top. Your difficulty about what you call 

 the "axial inclination " of tlie Moon, arises wholly from the fact 

 that her diurnal path is not parallel to the horizon. For a cognate 

 reason the Great Cross in the Constellation Cygnus rises in these 

 latitudes horizontally and sets perpendicularly. — J. Murray has 

 had an octagonal tin reflector made, and tinds that it gives a 

 number of dim blotches of light when a candle is placed 20 inches 

 from its centre. This, he seems to conceive, affords some sort of 

 explanation of Sunspots ! Like the lamented Artemus Ward, how- 

 ever, " I don't see where the larfture comes in myself." — Excelsior. 

 The Sidereal Messcn<ier is published at the Carleton College Obser- 

 vatory, Northfield, Minn., U.S. I do not know whether either of 

 the American publishers in London would be able to obtain a single 

 number for you. It is a subscription serial. — Mars. Xewcomb's 

 " Popular Astronomy " is an admirable book. By all means get 

 it. — Algernon Bbay. The words which you quote were used in 

 connection with the enormous amount of painstaking work 

 bestowed upon the production of the magic cube. I willingly 

 repeat them in connection with the specimen which you send me. 

 You will perhaps give me credit for knowing how much — or now 

 little — maihematiral difficulty is attendant on such calculations. 

 "X.'s" cube contains once each numberfroml to 1,331 inclusive. — 

 E. E. O'Callaghan asks me to correct the statement on p. 97 that 

 the National Food Reform Society is a Vegetarian Association ; inas- 

 much as it permits the nse of eggs, milk, cheese, and butter. 

 Possibly the reviewer may have been misled by the short manifesto 

 of the society, which heads all its publications : " National Food 

 Reform Society." — Objects : " To circulate useful information 

 regarding healthful, cheap, and natural articles of diet; to promote 

 thrift in food — temperance in eating — and the use of grains, fruits, 

 nuts, and other products of the vegetable kingdom as essential 

 articles of diet ; and to advocate abstinence from flesh of mammals, 

 birds, and fish;" and it may also be that the eggs, milk, cheese, 

 and butter, part of its programme, was not prominently insisted 

 upon (even if there was a single syllable about it) in the ''Maga- 

 zine" noticed. — E. F. H. To answer your queries in detail here 

 would be impossible. The pole star at the date of the erection of 

 the Great Pyramid was a Draconis, and nnt Orion, as you imagine. 

 The pole of the heavens in these latitudes is the point towards 

 which the northern end of the earth's axis is directed. Now, the 

 polar radius of the earth describes a cone in space in the course of 

 256948 years, or. putting it another way, the pole of the equator 

 revolves round the pole of the ecliptic in that time. Hence the 

 pole star for the time being will be that star most nearly vertical 

 over the north pole of the earth. To the rest of yonr questions you 

 will find categorical replies in my '' Universe of Stars," published 

 by Longmans & Co. I may add. by the bye, that the companion to 

 Polaris has preserved sensibly the same distance from its primary 

 ever since its discovery, but that it seems to be moving very slowly 

 round that primary in a direction opposite to that of the hands of a 

 watch. 



#ur iWatftrmatiral Column. 



NOTES ON EUCLID'S SECOND BOOK. 



By Richard A. Proctoe. 



Continued from p. 80.) 



Prof. II. — To produce a given ^ u C £ D 



strai'jhtUne (A B) so that the 



rectan(ile contained 6y the whole line thus produced and the given 



straight line may be equal to the square on the part produced. 



Produce A B to C and D, making B C equal to C D equal to A B. 

 Divide C D in E so that the rectangle C D, D E may be equal to the 

 square on C E. Then the rectangle A E, A B shall be equal to the 

 square on B E. 



For the square on B E is equal to the squares on B C, C E, 

 together with twice the rectangle BC, C E; that is, to the square 

 ou A B, the rectangle C D, D E (con^t), and twice the rectangle A B, 

 C E ; that is, to the rectangle contained by A B, and the line made 

 up of A B, D E, and twice C E. But the sum of these lines is 

 equal to A B, C D, and C E together, that is, to A B, B C, and C E, 

 or to A E. Hence the square on B E is equal to the rectangle A E, 

 AB. 



Props. 11 and II. correspond to the two solutions of the 

 quadratic 



a (a — x) — i' 

 which results as the analytical expression of the relation in 

 Prop. 11, when A B is made equal to a, and the smaller section of 

 A 15 equal to x. 



Props. 12 and 13 are important in solving geometrical problems 

 of a certain class, though Euclid himself makes no use of these 

 propositions. Each has a ijeneral and also an eiact converse 

 theorem. The general theorem converse to Prop. 12 is this : — If 

 the square on one side of a trianr/le is greater than the sum of the 

 squares on the other two sides, these two sides co7itain an ohtuse angle. 

 The proof is simple; the angle contained by the two sides must 

 either he acute, right, or obtuse. If it were acute, then by Euc. X., 

 Bk. II., 13, the stpiares on the sides containing this angle would 

 together be greater than the Sfiuare on the remaining side, but they 

 are not greater; if it were right, the squares on the sides con- 

 taining this angle would together be e(iual to the square on the 

 remaining side; but they are not equal to this square. Therefore 

 the angle is obtuse. 



And in like manner may be proved the theorem converse to Prop. 

 13, viz., — If the square on one side of a triangle he greiter than the 

 sum of the squares on the remaining sides these sides contain an 

 acute angle. 



These two props, may be re- 

 ferred to as Euc. II., Props. 12 

 and 13, gen. conv. 



The exact converse theorem 

 to Prop. 12 is this,— 7/ A C B be 

 an ohfuse angle, and B C 6e j>ro- 

 duced to D, so that the squares 

 on BC and AC with twice the 

 rectangle B C, C D are equal to 

 the square on AB, then AD is 

 perpendicular to B D. This property is often useful, as is the corre- 

 sponding property converse to Prop. 13, viz., — 7/ A B C be an acute 

 angle and a point D i;s taJcen in BC {produced if necessary), such 



that the squares on AH and BC together exceed the square on AC 

 hij twice the rectangle BC, BD, th&n AD is perpendicular to B D. 

 The proof in either case is easy, for in the first case, if the foot of 

 the perpendicular from A on B C produced, fell othervvise than at 

 D — at E suppose, it can be readily shown to follow from Prop. 12 

 that C D is equal to C E, which is absurd : and similarly in the 

 second case we can show (if E is the foot of the perpendicular 

 from A) that B E is equal to B D. 



These propositions mav be referred to as Euc. Bk. II., Props. 

 12, 13, c,v. com: 



