July 25, 1884.] 



• KNO^VLEDGE ♦ 



79 



absolutely unthinkable. It can bare noitlier centre nor circum- 

 ference. — W. Brooks Sayers. Would you be surprised to hear 

 that a gentleman named Aristarchus of Samos (who died B.C. 212) 

 anticipated yoa in your idea of determining the distance of the sun, 

 by observing the angle subtended by him and the dichotomised 

 moon, some 2,160 years ago? From the practical impossibility, 

 however, of ascertaining when the moon is exactly half-full, he 

 made out that the sun's distance from the earth was onh- nineteen 

 times that of the moon ; whereas, as we know, this is less than a 

 twentieth part of its true value. Your absurd endorsement of 

 " private " on your letter has delayed it exactly a week. — G.\ETn 

 WiLKixsoN opines that the vaccine eruption is contagious. — T. 

 Mann. The principal movements of the earth are : — 1. The 

 annual. 2. Her motion round the centre of gravity of herself and 

 the moon. 3. The diurnal. 4. What yon speak of as the pre- 

 cessional, or that conical motion of her axis which gives rise to 

 the Precession of the Equinoxes. 5. The Xutational. 6. The 

 eecular change in the obIi(]uity of the ecliptic. 7. The revolution 

 of her apsides. 8. The secular variation is the eccentricity of her 

 orbit, and, 9, her motion in space in common with the whole solar 

 system. The contributor to whom you refer seems to have forgotten 

 his promise. — A. JIackay. See reply to you on page Gl and para- 

 graph on page 02. — Jas. Gille.spie. Do, for goodness sake, master 

 the rudiments of your subject. Do you know that the earth's mean 

 equatorial semi-diameter only subtends an angle of 885" at the 

 Sun, and that the Pole Star iV< actually vertically overhead in 

 latitude 88° 11' north ? Polaris is not in the pole of the heavens, 

 but 1° 19' from it. You seem hopelessly incapable of comprehend- 

 ing that at the enormous distance of the fixed stars all lines 

 simultaneously drawn to them from any parts of the earth's 

 surface are parallel — nay, that so remote are they that even the 

 diameter of her orbit shrinks to a point as viewed from them ! 

 — J. Greevz Fi,*her sends me " Five Eulez for Improving 

 Spelling," for which I am much obliged ; but fancy that I shall 

 continue to employ, for some little time to come, orthography so 

 thoroughly " understanded of the people" as that which ho seeks 

 to supersede. — PoLY'GLOT. Thanks. Certainly not at present. — 

 Nigel Doble. Premising that ()6 inches is a ridiculously in- 

 adequate focal length for a 6-inch object-glass (which would per- 

 form infinitely better if it had a focus of 90 inches). I may tell 

 you that you would need at least four eye-pieces for astronomical 

 purposes, which, with a G6-inch objective, would be constructed as 

 follows, each, of course, consisting of two plano-convex lenses, with 

 the plane sides next the eye. So. 1 field-glass 1'65-inch focus, 

 eye-glass 0'55-inch focus, I'l inch apart. This would give a power 

 of 80. No. 2 field-glass, 0'73 inch focus, eye-glass 0'24 inch focus, 

 0'48 inch apart. This would magnify ISO diameters. No. 3 field- 

 glass, 0'53 inch focus, eye-glass 018 inch focus, 036 inch apart. 

 This would magnify 250 times. And No. 4 Field-lens, 026 inch 

 focus, and eye-lens 0'09 inch focus, 0'17 inch apart. This would 

 give a power of 500 for very close double stars. Each of these 

 eye-pieces should have a diaphragm, to limit the field of view, 

 placed in the focus of the eye-glass. I do not know the work you 

 mention. — Malcolm Fyte. The pernsal of your letter suggests the 

 dictum of the great American philosopher: "There's nothing new, 

 and there's nothing true, and it don't much signify." In everything 

 of the slightest value you have been anticipated by Mr. F. H. 

 Wenham and others — years ago. — Augustus J. Harvey suggests the 

 appointment of a staff of commissionaires to conduct visitors to 

 the principal points of interest of the Health Exhibition, and to 

 explain the most important objects shown there. — The President 

 AND Council of the Society for the Study and Cure of Inebriety. 

 I regret that I was a good many miles from London on the evening 

 of the 21st. — H. W. Jackson. Please communicate with Messrs. 

 Wyman & Sons, the publishers of Knowledge. — Law. Prior to the 

 receipt of your letter I was ignorant that there were any Whitworth 

 Law Scholarships at all. — W. E. Snell. No one, that I am aware 

 of, has ever disputed in these columns the value of sanitation as one 

 form of prevention of small-pox. — A. McDonnell. I will read your 

 pamphlet as impartially as I can. — Eye-witness. Something of the 

 sort was attempted on p. 247 of our last volume, but did not seem 

 to meet any general want. My large star-atlas would enable you 

 to learn the constellations. — G. Pixnington. " I think not," says 

 the lamb. — G. C. I do not know the particular book you mention, 

 but the names of its joint authors afford a sutBcient guarantee of 

 its excellence. Bloxam's is my text-book. Churchills publish it, 

 and it is first-class, but it cost rather more than your specified 

 price. 



In response to numerous applications, it has been decided 

 to issue "The Index " to Volume V. of Knowledge, ivith 

 the present numier,/ree of charge. 



(Bur iHattjfmati'ral Column. 



n 



NOTES ON EUCLID'S SECOND BOOK. 

 By Richard A. Proctor. 

 {Continued from p. 41.) 



IT is well to notice the algebraical and arithmetical relations 

 which the different relations presented in the preceding proposi- 

 tions serve to illustrate. 



We must show first that if each of the two lines which contain 

 a rectangle can be divided into an exact number of parts each equal 

 to some unit of linear measurement, then the product of the two 

 numbers represents the number of corresponding units of square 

 measurement contained in the rectangle. 



Let the rectangle A B C D be contained by the lines A B, AD; 

 and suppose that a 

 certain unit of length '^ " 

 is contained 13 times 

 in A B and 7 times 

 in AD. Then if A B 

 be diWded into 13 

 equal parts and A D 

 into 7 equal parts, 

 each part of each 

 line is equal to this 

 um't of length. And 

 if we draw through the points of division in A B lines parallel to 

 AD, and through the points of division in AD lines parallel to 

 A B, it is clear that the rectangle A B C D will be divided into a 

 number of squares each having its sides equal to the unit of len<'th. 

 Now each row of squares parallel to A B contains 13 such squares, 

 and there are seven such rows. Therefore the -whole rectangle 

 contains 7 times 13 squares. Thus the product of the numbers 7 

 and 13, which represent the length of the sides in terms of the 

 linear unit, gives us the number representing the area of the rect- 

 angle in terms of the corresponding unit of square measurement. 

 And the proof would have been precisely the same whatever the 

 number of units of linear measurement in the sides A B, AD, — so 

 that, if A B contains a such units, and A D contains b, the 

 rectangle A B C D contains a b units of square measurement. 



It would be easj' to extend this proof to the case of a rectangle 

 having incommensurable sides ; but for the purpose of illustration 

 the case of commensurable sides is sutficient. This commenstira- 

 bility is to be understood as implied in what follows. 



In Euc, Book II., Prop 1, if the undivided line contain a units 

 of length, the several parts of the divided line b, c, and d units, 

 respectively, the proposition corresponds to the algebraical 

 identity 



a(b + c + d) =ab + ac + ad. 



Prop. 2. — If the undivided line contain (a + b) units of length, 

 its parts a and h units correspond to the identity 

 (a + b)a+ (a + h)b={a + by-' 



In Prop. 3, on the same supposition, the algebraical identity 

 corresponding to the proposition is 



{a + b)b = ab + b'. 



Prop. 4, on the same supposition, corresponds to the identity 

 {a + b)-' = a^ + 2ab + b' 



In Prop. 5, let AB = 2a, and CD = 6, so that AD = (a-l-6) and 

 D B = (a — fc) then the corresponding algebraical identity is 



(a + b) (a-b) +b' = a' 

 that is, the well known relation 



a-—b-= (a + b) (a — b). 



But if we put A D = a and D B = b, we obtain the relation — 



(a — b\- /a + b \- 

 — )=(-!-) = 

 that is, the well known formula — 



{a + by- (a-by=iab 

 We get the same identities in the case of Prop. 6, if we make 

 corresponding suppositions, simply interchanging a and b. 



In Prop. 7, put AC = a, and BC = b, then the algebraical 

 identitv corresponding to the proposition is 



(a¥by'-\-b'' = a- + 2b{a + b) 

 In Prop. 8 put AC = a, and BC = 5; then the corresponding 

 algebraical relation is 



4(o-t-b)b-Ha= = (a-H2b)2. 

 In Prop. 9 put first A B = 2 a and C D = b ; then the corresponding 

 algebraical identity is 



(a-f b)'-f (a-5)- = 2(a--Hb'). 

 Nest put A D = a and D B = b, and we obtain the relation 



„= + b= = 2(«-±-7.2(«-rL^y 



