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former is greater than, equal to, or less than the angle opposite sible, let the straight line a x be drawn meeting B C and equal to the latter. This corollary was, by mistake, again appended to A E. Then, because a H is equal to A B, as just proved, and to Prop. XVIII. in our last lesson; whereas, the following Ax is by hypothesis equal to A È, therefore a x is equal to AH, corollary should have been appended to it:-One angle of a by Axiom 1. ; but it has been proved that a H, a straight line triangle is greater than, equal or less than another, accord- nearer to the perpendicular AD is always less than ax, a ing as the side opposite to the former is greater than, equal to, straight line more remote from it; therefore, the straight line or less than the side opposite to the latter.
A K is both equal to ah and less than it, which is impossible ; Corollary 2.-All the angles of a scalene triangle are un- wherefore a k is not equal to AB; and in the same manner it equal.
can be proved that no other straight line than a h can be equal EXÉRCISE TO PROPOSITION XIX.
Wherefore from the same point A, only two equal If from a point without a given straight line, any number of straight lines an and Ae can be drawn to the given straight straight lines be drawn to meet it ; of all these straight lines, that Therefore, if from a point without a given straight line, &c.
line Bc, one upon each side of the least straight line A D. which is perpendicular to the given straight line is the least; and of
Q.E.D. others, that which is nearer to the perpendicular is always less* than the more remote ; also from the same point only two equal
PROPOSITION XX.-THEOREM. straight lines can be drawn to the given straight line, one upon each side of the least straight line.
Any two sides of a triangle are together greater than the third In fig. v, let a be the point, and b c the given straight line; side. also let any number of straight lines AD, A E, A P, and ag, be
In fig. 20, let ABC be a triangle;
any two of its sides are together
First, to prove that the two sides BA and A C are together greater than BC. Produce B A to the point D, and make an equal to A o by Prop. III. Join D c.
Because, in the triangle a DC, the side DA is, by construcBA
tion, equal to the side Ac; therefore the angle adc is equal to the angle A CD. But the angle BCD, by Axiom IX., is greater than the angle A OD; therefore, the angle BCD is also
Because the angle drawn from the point A to meet the straight line B C, and let BCD, of the triangle BCD, is greater than its angle B D C, and
greater than the angle ADC, or BDC. AD be perpendicular to bc, Prop. XII.; then, of all these the greater angle is subtended by the greater side, Prop. XIX., straight lines A D is the least, and of the others, Ag is less than therefore the side B D is greater than the side Bc. Again, in AF, and a r than AG; also from the point a, only two equal the triangle ADC, the side a d is equal to the side A c, by constraight lines can be drawn to the straight line BC, one upon struction ; to each of these equals, add the side Ba; then, each side of A D.
BD, the whole side of the triangle BCD, is equal to the two Because the straight line a d is by hypothesis perpendicular sides BA and ag of the triangle BAC. But the side B D of the to the straight line BC, therefore, by Def. 10, each of the triangle BCD, was proved to be greater than its side Bo; thereangles A De and Adi is a right angle; and they are, there fore, the two sides BA and A c of the triangle B A C are together fore, by Axiom XI., equal to one another; but the exterior greater than its third side Bc. In the same manner, it may be angle A D u of the triangle AdB, is, by Prop. XVI., greater proved that the two sides A B and B C are together greater than than its interior and remote angle A Ed, therefore, also the Ac; and the two sides Bc and c A are together greater than angle Ade is greater than the angle AED; wherefore, by AB. Therefore, any two sides of a triangle, &c. Q. E. D. Prop. XIX., the side A E is greater that the side a D. In the same manner, it may be shown that the straight lines makes the following proper remarks :-“Proclus, in his com
Scholium.-Dr. Simson, in his notes to his edition of Euclid, AF and AG are also greater than the straight line A D. Wherefore of all the straight lines A D, A E, AP, and Ad, the mentary (on Euclid), relates, that the Epicureans
derided perpendicular À D is the least.
Prop. XX. as being manifest even to asses, and needing no Next, because the exterior angle Ad # of the triangle App it be manifest to our senses, yet it is science which must give
demonstration ; and his answer is, that though the truth of is by Prop. XVI. greater than its interior and remote angle the reason why two sides of a triangle are greater than the AFD, therefore also the angle a dris greater than the angle afd. third; but the right answer to this objection against this and Again, because the angle ADF has een shown to be greater some other propositions is, that the number of axioms ought than the angle AfD or A FË, and that the exterior angle A E F of the triangle A D E is greater by Prop. XVI. than its interior not to be increased without necessity, as it must be, if these angle A D B, much more, therefore, is the angle a ef greater XX. is merely a corollary to the definition of a straight line
propositions be not demonstrated.". It is true that this Prop. than the angle a pe; wherefore, in the triangle A Pr, by Prop. given by Archimedes, namely, that “it is the shortest disXVIII., the side A p is greater than the side A e. In the same manner, it may be proved that the straight line Ad is greater the two points B and c, taken along the straight line, is
tance between any two points ;” for the distance between than the straight line A F. Therefore, of the other straight evidently less than the distance between these
points taken lines, AB is less than af, and a r than a g; that is, the straight along the crooked line BAC; and as even asses or drunken line nearer the perpendicular is always less than the more
men endeavour to take the shortest road to their desired remote. to DĒ, by Prop III., and join A R. Because in the two triangles laugh and mock at everything that did not just exactly square Lastly, from the straight line pc, cut off the part da equal object, there seems to be some foundation for the derision of
the Epicureans; but these philosophers were accustomed to ADE and AD, the two sides a p and d i are equal to the with their views; hence they said even of the great' Apostle two sides a D and D e, and the angle
AD H is equal to the angle Paul, when preaching Jesus and the resurrection at Athens : AD E, therefore, by Prop. IV., the base a n is equal to the base - What will this babbler say?" Hence, it is evident, that if A B; and no other straight line equal to A r, but an, can be Paul had given them a mathematical demonstration of the drawn from the point i to the straight line Bc. For, if pos- resurrection of the dead, they would not have believed him,
• By mistake printed greater, in the earlier editions of Cangell's but would have continued to mock on, like infidels in modern Euclid.
times. Now, they have both Moses and the prophets, and This exercise was solved by T. Bocock, Great Warley ; Quintin Christ and his apostles, and if they believe not
them, neither Pringle, Glasgow; J. H. Eastwood, Middleton and others,
would they believe if one rose from the dead.
Schotium 2.—This proposition may be demonstrated by XV.; therefore, the angle arg is equal to the angle PEG, by another method, as follows :- In fig.
Axiom I. But the side A E is equal to the side E , by construc. v, let B A C be a triangle, and let it be Fig. V.
Fig. X. required to prove that the two sides BA and A c are greater than the third side Bc. Bisect the angle BAC by the straight line ad, meeting Bc in D, by Prop. IX. Then, because the exterior angle BDA of the triangle DAO is greater than its interior and remote angle D A C, by Prop. XVI., and the exterior angle CDA of the triangle BDA is greater than its interior and remote angle DAB; and that the angles DAC and D A B are equal; therefore, the angle B Da is also greater than the angle DA B, and the angle cd A than the angle DAO; wherefore, by Prop. XIX., the side ba is greater than B D, and the side ca greater than cd; therefore the two sides BA and A c are greater than the whole side BC.
EXERCISE I. TO PROPOSITION XIX.
Any side of a triangle is greater than the difference between the other two sides. If the triangle be equilateral, the truth of the proposition is tion, and the side e G is common to the two triangles A BG
evident; for the difference between and PBG; therefore, the base GF is equal to the base A G. In Fig. w.
any two sides is nothing. If the the same manner, it may be shown that the straight line fu is triangle be isosceles, the base or equal to the straight line A H. Because the straight line A G is third of the triangle is greater than equal to the straight line GF, if to these equals we add the the difference between the other two straight line GB, the two straight lines a G and a B are equal to sides, which are equal, for the same the whole FB. But the two sides pu and 1 B, of the triangle
FH B, are together greater than the side F B; and as A u is In any other case, let B A 1, fig. w, equal to Fr, therefore au and us are together greater than be a triangle, of which the side B o'is greater than the side BA PB. But it has been shown that A G and & B are equal to FB; then the remaining side a c is greater than the difference be thefore a H and H B are together greater than A G and G B, that tween the other two sides, BC and B A.
is, AG and G B are together less than the sum of A 1 and 1 B. From BC, the greater side, cut off B D, a part equal to the And the same may be proved of the two straight lines drawn less side B a, by Prop. III.
from the points A and B to any other point in the straight line Because the two sides B A and a c are together greater than CD. Therefore, from two given points A and B on the same BC, by Prop. XX., and that BD is, by construction, equal to side of the straight line CD, iwo straight lines have been drawn BA; therefore, taking these equals away, the remainder Ac is to a point a in it, which taken together are less than the sum greater than the remainder DC. Therefore, any side of a tri. of two straight lines drawn from the same points to any other angle, &c. Q. E. D.
point in od. Q. E. F.
Scholium 2. In the preceding, demonstrations, it is very EXERCISE II, TO PROPOSITION XX.
properly remarked by T. Bocock, Great Warley, that this The three sides of a triangle taken together are grealer than the another, and if the same or equal magnitudes be added to each,
axiom is taken for granted, viz. "If one magnitude be greater double of any one side, but less than the double of any two sides.
the same inequality will remain; that is, the sum of the Because any two sides of a triangle are greater than the greater magnitude and that which is added to it will be third side, by Prop. XX.; therefore, if the third side be added to greater than the sum of the less magnitude and that which is these unequals, the three sides taken together are greater than added to it.” Another axiom is also taken for granted, viz., twice the third side. Again, because one side of a triangle is “If one magnitude be greater than another, and if the same less than the other two sides, by Prop. XX., therefore, if the or equal magnitudes be taken from each, the same inequality other two sides be added to these unequals, the three sides will remain ; that is, the difference between the greater magtaken together are less than twice the other two sides. There- nitude and that which is taken from it, will be greater than fore, the three sides, &c., Q. E. D,
the difference between the less magnitude and that which is
taken from it." EXERCISE III, TO PROPOSITION XX.
* The exercises on Problem XX., were solved as follows: I., II. and From two given points on the same side of a straight line given u. by J. H. Eastwood, Middleton; E. J. Bremner, Carlisle ; T. in position, to draw two straight lines which shall meet at a point Bocock, Great Warley ; Quintin Pringle, Glasgow; C. L. Hadfield in it, and which taken together shall be less than the sum of two and J. Goodfellow, Bolton-le-Moors; I. and III. by E. ... Jones, Pemstraight lines drawn from the same points to any other point in the broke Dock; I. and II. by E. Ru88, Pentonville ; and I. by J. Jenkins, given straight line.
Pembroke Dock. In fig. x, let A and B be the two given points, and cd the straight line given in position. From the points a and B it is required to draw two straight lines which shall meet at a
ANSWERS TO CORRESPONDENTS. point in the straight line cd, and which taken together shall this journal, natural faith we dont understand, and the only book, op
E. WILKINSON (York): We eschew politics, and all mention of them in be less than the sum of two straight lines drawn from the Christian fain is the BIBLE.-JOHN HERN : Yes.-T. O. (Hainsworth): points A and B, to any other point in the straight line CD. Very good.-A WELSHMAN
(of Anglesea) was answered before. It is all From the point a, draw the straight line a E at right angles horse power; he must just eat what is good for him, and this he can only
stuff about physiology and food ; man is not a steam-engine of a certain to the straight line cd, by Prop. XI., and produce it to the ind out by experience.—YOUNG NATURALIST: We don't know.-GERMANI. point r, making the part BP equal to the part A E, by Prop. cus (Edinburgh), T. C. W., (London), and X. ¥. 2. (Dublin): Yes.-T. fil. Join FB, and let it cut o p in the point G. Join Ag. SHEPHERD (Salford) and J. FARNDON (Birmingham): Thanks.-W. Then the straight lines A G and a B drawn from the points A and Thanks.-FAIR PLAY (Waterford) and LOUIS LE PLUS JEUNB: We don't
WALKER, ada Trou 32; Right.-AMATOR SOIENTIAE (Fenchurch-street): B and meeting cd in o, are the two straight lines required. know.-E. MORRIS: Write to Mr. Bell.--A SUBSCRIBER (Colne): RightIn CD. take away any other point n, and join a 1 and BH, STUDETE (Hampstead-road), should call on Henry Moore, Èsq., Secretary to Because the angle A e G is equal to the angle Aeo by con- Greek Seriptures, apply to Messrs. Bagster, Paternoster-row.-8. F. HEN
the University, Somerset House, for a solation of all his queries. For the struction, and the angle A e c equal to the angle reo by Prop. I Best (Fordingbridge) : Received.
LESSONS IN BOOKKEEPING.--No. X I.
(Continued from page 177).
T&B Journal, as we have before remarked, is no longer what I has with the Ledger, we mean the GENERAL POSTING Book. its name denotes, a Day Book; but is now used, in Double Some of our students who are, no doubt, keen business-men, Entry, as a book for collecting all the transactions of business and are on the alert to discover any improvements that can be for a given period into a focus, previous to their being entered made in Bookkeeping, in order to shorter their labour, and in the Ledger. In an ordinary business, where the transactions produce more accurate results, or, rather, to effect less frequent are neither too numerous nor too complicated, the formation liability to error, will, if they have gone with us thus far, proof this book from the various subsidiary books of the concern, pose some shorter or more pointed name than the precedings may take place only once a month; and then with reference to for once, therefore, we leave this subject in their hands. Ali time, as we formerly observed, it might be called the Month- we shall say, is this : that gentlemen who have been in busi. Book ; and in the same way, according to the regular intervals ness for twenty, thirty, aye, and forty years, have thanked us when this collective book is made up, it might be called Week- personally many times for the lessons on this subject which Book, or even Day-Book. The best name, however, which they have received from us, and particularly in reference to could be given to it, would be one indicative of its actual use, our method of striking a General Balance, exemplified at the without reference to time; we have already suggested the end of this Journal, but which cannot be fully explained in this name Sub-Ledger, and we may now propose a name which lesson, as the Trial Balance and Ledger have not yet been subwould, perhaps, be more accurate and distinct, as regards the mitted to the student. This will be done in our next lesson. method in which it is made up, and the connexion which it