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former is greater than, equal to, or less than the angle opposite to the latter. This corollary was, by mistake, again appended to Prop. XVIII. in our last lesson; whereas, the following corollary should have been appended to it:-One angle of a triangle is greater than, equal to, or less than another, according as the side opposite to the former is greater than, equal to, or less than the side opposite to the latter. Corollary 2.-All the angles of a scalene triangle are unequal.

EXERCISE TO PROPOSITION XIX.

If from a point without a given straight line, any number of straight lines be drawn to meet it; of all these straight lines, that which is perpendicular to the given straight line is the least; and of others, that which is nearer to the perpendicular is always less* than the more remote; also from the same point only two equal straight lines can be drawn to the given straight line, one upon each side of the least straight line.

In fig. u, let A be the point, and Bc the given straight line; also let any number of straight lines AD, A E, A F, and ▲ G, be

drawn from the point A to meet the straight line BC, and let AD be perpendicular to BC, Prop. XII.; then, of all these straight lines A D is the least, and of the others, AE is less than AF, and AF than AG; also from the point A, only two equal straight lines can be drawn to the straight line в C, one upon each side of A D.

Because the straight line A D is by hypothesis perpendicular to the straight line BC, therefore, by Def. 10, each of the angles A D E and ADH is a right angle; and they are, therefore, by Axiom XI., equal to one another; but the exterior angle ADH of the triangle ADE, is, by Prop. XVI., greater than its interior and remote angle A ED, therefore, also the angle ADE is greater than the angle ARD; wherefore, by Prop. XIX., the side AE is greater that the side A D. In the same manner, it may be shown that the straight lines AF and AG are also greater than the straight line AD. Wherefore of all the straight lines AD, A E, AF, and AG, the perpendicular AD is the least.

Next, because the exterior angle A D H of the triangle AFD is by Prop. XVI. greater than its interior and remote angle AFD, therefore also the angle ADF is greater than the angle AF D. Again, because the angle ADF has been shown to be greater than the angle AFD or AFE, and that the exterior angle A EF of the triangle A D E is greater by Prop. XVI. than its interior angle A DE, much more, therefore, is the angle AEF greater than the angle AFE; wherefore, in the triangle A EF, by Prop. XVIII., the side AF is greater than the side A E. In the same manner, it may be proved that the straight line AG is greater than the straight line AF. Therefore, of the other straight lines, AE is less than AF, and AF than AG; that is, the straight line nearer the perpendicular is always less than the more Lastly, from the straight line DC, cut off the part DH equal to DE, by Prop III., and join A. Because in the two triangles ADE and ADH, the two sides a p and DH are equal to the two sides A D and D E, and the angle ADH is equal to the angle ADE, therefore, by Prop. IV., the base a H is equal to the base AE; and no other straight line equal to AF, but AH, can be drawn from the point A to the straight line BC. For, if posBy mistake printed greater, in the earlier editions of Cassell's This exercise was solved by T. Bocock, Great Warley; Quintin Pringle, Glasgow; J. H. Eastwood, Middleton and others.

remote.

Euclid.

sible, let the straight line AK be drawn meeting B c and equal to A E. Then, because a н is equal to AE, as just proved, and AK is by hypothesis equal to ▲ E, therefore A K is equal to A H, by Axiom I.; but it has been proved that A н, a straight line nearer to the perpendicular AD is always less than AK, a straight line more remote from it; therefore, the straight line AK is both equal to AH and less than it, which is impossible; wherefore AK is not equal to AE; and in the same manner it can be proved that no other straight line than AH can be equal to A E. Wherefore from the same point A, only two equal straight lines AH and AE can be drawn to the given straight line BC, one upon each side of the least straight line AD. Therefore, if from a point without a given straight line, &c.

Q. E. D.

PROPOSITION XX.-THEOREM.

Any two sides of a triangle are together greater than the third side.

In fig. 20, let ABC be a triangle; any two of its sides are together greater than the third side; viz., the two sides BA and AE are together greater than BC; the two sides AB and BC are together greater than Ac; and the two sides BC and CA are together greater than A B.

First, to prove that the two sides

Fig. 20.

BA and A c are together greater than B C. Produce BA to the point D, and make A D equal to a c by Prop. III. Join D C.

Because, in the triangle A D C, the side DA is, by construction, equal to the side A c; therefore the angle ADC is equal to the angle AC D. But the angle B CD, by Axiom IX., is greater than the angle A OD; therefore, the angle BCD is also greater than the angle ADC, or BD C. Because the angle BCD, of the triangle B CD, is greater than its angle B D C, and the greater angle is subtended by the greater side, Prop. XIX., therefore the side B D is greater than the side BC. Again, in the triangle ADC, the side A D is equal to the side a c, by construction; to each of these equals, add the side BA; then, BD, the whole side of the triangle BCD, is equal to the two sides BA and AC of the triangle BAC. But the side B D of the triangle B CD, was proved to be greater than its side Bo; therefore, the two sides BA and A C of the triangle B A C are together greater than its third side BC. In the same manner, it may be proved that the two sides A B and B C are together greater than Ac; and the two sides BC and CA are together greater than AB. Therefore, any two sides of a triangle, &c. Q. E. D.

Scholium.-Dr. Simson, in his notes to his edition of Euclid, makes the following proper remarks:-"Proclus, in his com mentary [on Euclid], relates, that the Epicureans derided Prop. XX. as being manifest even to asses, and needing no it be manifest to our senses, yet it is science which must give demonstration; and his answer is, that though the truth of the reason why two sides of a triangle are greater than the third; but the right answer to this objection against this and some other propositions is, that the number of axioms ought not to be increased without necessity, as it must be, if these XX. is merely a corollary to the definition of a straight line propositions be not demonstrated." It is true that this Prop. given by Archimedes, namely, that "it is the shortest distance between any two points;" for the distance between the two points B and C, taken along the straight line, is evidently less than the distance between these points taken along the crooked line BAC; and as even asses or drunken

men endeavour to take the shortest road to their desired

object, there seems to be some foundation for the derision of laugh and mock at everything that did not just exactly square the Epicureans; but these philosophers were accustomed to with their views; hence they said even of the great Apostle Paul, when preaching Jesus and the resurrection at Athens:

What will this babbler say?" Hence, it is evident, that if Paul had given them a mathematical demonstration of the resurrection of the dead, they would not have believed him, but would have continued to mock on, like infidels in modern times. Now, they have both Moses and the prophets, and Christ and his apostles, and if they believe not them, neither would they believe if one rose from the dead.

Fig. V.

Schotium 2.-This proposition may be demonstrated by another method, as follows:-In fig. v, let B A C be a triangle, and let it be required to prove that the two sides BA and A C are greater than the third side BC. Bisect the angle B A C by the straight line AD, meeting BC in D, by Prop. IX. Then, because the exterior angle BDA of the triangle DAC 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 DA B are equal; therefore, the angle B DA is also greater than the angle DA B, and the angle CDA than the angle DAC; 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 AC are greater than the whole side BC.

reason.

In any other case, let B A , fig. w, be a triangle, of which the side B O is greater than the side BA; then the remaining side a c is greater than the difference between the other two sides, BC and B A.

From BC, the greater side, cut off BD, a part equal to the less side BA, by Prop. III.

Because the two sides BA and AC are together greater than BC, by Prop. XX., and that BD is, by construction, equal to BA; therefore, taking these equals away, the remainder A c is greater than the remainder DC. Therefore, any side of a triangle, &c. Q. E. D.

EXERCISE II. TO PROPOSITION XX.

The three sides of a triangle taken together are greater than the double of any one side, but less than the double of any two sides.

Because any two sides of a triangle are greater than the third side, by Prop. XX.; therefore, if the third side be added to these unequals, the three sides taken together are greater than twice the third side. Again, because one side of a triangle is less than the other two sides, by Prop. XX., therefore, if the other two sides be added to these unequals, the three sides taken together are less than twice the other two sides. Therefore, the three sides, &c., Q. E. D,

EXERCISE III, TO PROPOSITION XX.

From two given points on the same side of a straight line given in position, to draw two straight lines which shall meet at a point in it, and which taken together shall be less than the sum of two straight lines drawn from the same points to any other point in the given straight line.

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 point in the straight line CD, and which taken together shall be less than the sum of two straight lines drawn from the points A and B, to any other point in the straight line CD.

tion, and the side E G is common to the two triangles A EG and FEG; therefore, the base GF is equal to the base A G. In the same manner, it may be shown that the straight line FH is equal to the straight line A H. Because the straight line AG is equal to the straight line GF, if to these equals we add the straight line GB, the two straight lines A G and G B are equal to the whole FB. But the two sides FH and HB, of the triangle FHB, are together greater than the side FB; and as A His equal to FH, therefore AH and HB are together greater than But it has been shown that A G and G B are equal to FB; thefore ▲ H and H в are together greater than A G and GB, that is, AG and GB are together less than the sum of А H and H B. And the same may be proved of the two straight lines drawn from the points A and B to any other point in the straight line CD. Therefore, from two given points A and B on the same side of the straight line CD, two straight lines have been drawn to a point & in it, which taken together are less than the sum of two straight lines drawn from the same points to any other point in c D. Q. E. F.

FB.

Scholium 2. In the preceding demonstrations, it is very properly remarked by T. Bocock, Great Warley, that this another, and if the same or equal magnitudes be added to each, axiom is taken for granted, viz. "If one magnitude be greater the same inequality will remain; that is, the sum of the greater magnitude and that which is added to it will be greater than the sum of the less magnitude and that which is added to it." Another axiom is also taken for granted, viz., "If one magnitude be greater than another, and if the same or equal magnitudes be taken from each, the same inequality will remain; that is, the difference between the greater magnitude and that which is taken from it, will be greater than the difference between the less magnitude and that which is taken from it."*

III. by J. H. Eastwood, Middleton; E. J. Bremner, Carlisle; T *The exercises on Problem XX., were solved as follows: I., II, and Bocock, Great Warley; Quintin Pringle, Glasgow; C. L. Hadfield and J. Goodfellow, Bolton-le-Moors; I. and III. by E. L. Jones, Pembroke Dock; I. and II. by E. Russ, Pentonville; and I. by J. Jenkins, Pembroke Dock.

ANSWERS TO CORRESPONDENTS.

this journal; natural faith we dont understand, and the only book of E. WILKINSON (York): We eschew politics, and all mention of them in Christian faith is the BIBLE.-JOHN HEBN: Yes.-T. O. (Hainsworth): Very good.-A WELSHMAN (of Anglesea) was answered before. It is all 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 find out by experience.-YOUNG NATURALIST: We don't know.-GERMANIcus (Edinburgh), T. C. W. (London), and X. Y. Z. (Dublin): Yes.-T. Join AG. SHEPHERD (Salford) and J. FARNDON (Birmingham): Thanks.-W. Thanks.-FAIR PLAY (Waterford) and LOUIS LE PLUS JEUNE: We don't WALKER, and ТLOH 32; Right.-AMATOR SCIENTIAE (Fenchurch-street): know.-E. MORRIS: Write to Mr. Bell.-A SUBSCRIBER (Colne): RightSTUDETE (Hampstead-road), should call on Henry Moore, Esq., Secretary to the University, Somerset House, for a solution of all his queries. For the BEST (Fordingbridge): Received. Greek Scriptures, apply to Messrs. Bagster, Paternoster-row.-S. F. HEN

From the point a, draw the straight line AE at right angles to the straight line CD, by Prop. XI., and produce it to the point F, making the part EF equal to the part AE, by Prop. III. Join FB, and let it cut CD in the point G. Then the straight lines A G and GB drawn from the points A and B and meeting CD in G, are the two straight lines required. In CD. take away any other point н, and join a н and в н. Because the angle AEG is equal to the angle ABC by construction, and the angle A E C equal to the angle FEG by Prop.

LESSONS IN BOOKKEEPING.-No. XI.

THE JOURNAL.

(Continued from page 177).

THE Journal, as we have before remarked, is no longer what its name denotes, a Day Book; but is now used, in Double | Entry, as a book for collecting all the transactions of business for a given period into a focus, previous to their being entered in the Ledger. In an ordinary business, where the transactions are neither too numerous nor too complicated, the formation | of this book from the various subsidiary books of the concern, may take place only once a month; and then with reference to time, as we formerly observed, it might be called the MonthBook; and in the same way, according to the regular intervals when this collective book is made up, it might be called WeekBook, or even Day-Book. The best name, however, which could be given to it, would be one indicative of its actual use, without reference to time; we have already suggested the name Sub-Ledger, and we may now propose a name which would, perhaps, be more accurate and distinct, as regards the method in which it is made up, and the connexion which it

(1)

Date. Fol.

Cash Account Dr.

has with the Ledger, we mean the GENERAL POSTING BOOK. Some of our students who are, no doubt, keen business-men, and are on the alert to discover any improvements that can be made in Bookkeeping, in order to shorten their labour, and produce more accurate results, or, rather, to effect less frequent liability to error, will, if they have gone with us thus far, propose some shorter or more pointed name than the precedings for once, therefore, we leave this subject in their hands. All we shall say, is this: that gentlemen who have been in business for twenty, thirty, aye, and forty years, have thanked us personally many times for the lessons on this subject which they have received from us, and particularly in reference to our method of striking a General Balance, exemplified at the end of this Journal, but which cannot be fully explained in this lesson, as the Trial Balance and Ledger have not yet been submitted to the student. This will be done in our next lesson,

JOURNAL.

January, 1853.

To Sundries, as per C. B. fol. 1