ticularly when it denotes descent or children; e. g. Gian-nuôl đi Se-ce-gi-no, Cle-co di Mes-sê-re An-giu-liê-ri, in Boccaccio, where fi-glius-lo, child or son, is understood. EXERCISES.-ENGLISH-ITALIAN.* The rising of the sun. The dawn of the day. The return of spring. The warmth of the air. The beauty of the flower. The darkness of the night. The abyss of error. The fertility of the fields. The colours of the rainbow. The senses of man. The faults of young men. Money is the soul of commerce. Usage is the legislator of languages. The master of the garden is not here. The palace belongs to the prince. Here are the rooms of the uncle. The dresses belong to the cousin and not to the aunt. The brother tells the sister the will of the father. The children must always obey the parents. The physicians say the disorder shortens life. Exercise is useful to the body and to the mind. The countenance is the mirror of the soul. Tranquillity of mind is the highest degree of happiness. Temperance is the treasure of the wise man. The true ornament of the soldier is courage. The practice leads to perfection. Interest, pleasure, and glory, are the three motives of the actions and of the behaviour of men. Dawn, spun-tár, m. Day, giór-no, m. Spring, pri-ma-vê-ra, f. Beauty, bel-léz-za (ts), f. Error, er-ró-re, m. Fertility, fer-ti-li-tà, f. VOCABULARY, Rainbow, ar-co-ba-lé-no, m. Man, uô-mo, m. Fault, er ró-re, m. Young man, gió-va-ne, m. Soul, d-ni-ma, f. Commerce, com-mêr-cio, m. Is, è Legislator, le-gi-sla tire, m. Uncle, zi-o (ts), m. Dress, d-bi-to, m. Belong, ap par-tên-go-no Cousin, cu-gi-na, f. And not, è non Aunt, zi-a (ts), f.. Sister, so-rêl-la, f. Will, vo-lon-tà, f. Must always obey, de-vo-no Disorder, dis-or'-di-ne, m. Exercise, mô-to, m. Is useful, gió- và Body, côr-po, m. Mind, spi-ri-to, m. Is, è Mirror, spec-chio, m. EXERCISES.-ITALIAN-ENGLISH. Al-le La me-mô-ria. Dél-la ciê-ra. Al-la col-lí-na. Dál-la spia-ná-ta. Le bec-che-rí-e. Dél-le lo-can-de. Al-le pôrte. Dál-le strá-de. In fac-cia. Nél-la ví-gna. Nél-le fo-réste. Con pá-glia. Cól-la ví-te. Cól-le pén-ne. Per disgrá-zia. Per la vál-le Per le scioc-chéz-ze. Súl-la car-rôzza. Súl-le rú-pi. L'au-rô-ra. Dell' al-le-gréz-za. All' opi-nió-ne. Dall' o-ste-rí-a. Le i-dê-e. Dell' êr-be. ar-ti. Dál-le cit-tà. In i-slit-te. Nell' im-ma-gi-na-zió-ne. Nél-le a-ni-me. Con á-cqua. Coll' ún-ghia. Cól-le in-ségne. Per a-mi-ci-zia. Per l' as-si-cu-ra-zió-ne. Per le azió-ni. Sull' in-sa-la-ta. Súl-le in-fer-riá-te. Un fan-ciúllo. U-no stól-to. Un a-ni-má-le. U-na set-ti-má-na. D'un fiú-me. Ad ú-no schiop-pet-tiê-re. Da ú-na bal-le-ri-na. In ú-na chiê-sa. Con un ba-stó-ne. Per ú-no sco-lá-re. Su d' un sas-so, só-pra un sás-s0. VOCABULARY. Per, for, through, on account Countenance, fi-so-no mi-a, f. Disgrazia, misfortune, dis Soul, d-ni-ma, f. Tranquillity, quiê-te, f. Highest degree, cól-mo, m. grace (per dis-gra-zia, unfor- Valle, valley. Carrozza (ts), carriage, coach. Aurora, aurora, dawn. Allegrezza (ts), joy. Opinione, opinion. Arte, art. Città, town, city (no change in the pl.) Slitta, sledge. Immaginazione, imagination. Anima; mind, soul. Acqua, water. | Unghia, nail. Insegna, sign, arms, colours. Assicurazione (ts), security, in surance. Azione, action. Insalata, salad. Inferriata, iron-grate. Stolto, fool. Animale, animal. | Settimana, week. Fiume, river. Schioppettiere, arquebusier, Ballerina (f.), dancer. Bastone, stick. Scolare, pupil. Su, sopra, upon. LESSONS IN GEOMETRY.-No. XXIV. LECTURES ON EUCLID. (Contiuued from page 50). BOOK I.-PROPOSITION XIX.-THEOREM. The greater angle of every triangle is subtended by the greater side (that is, has the greater side opposite to it). In fig. 19, let ABC be a triangle hav- Three motives, tre mo-ti-vi, pl. opposite the angle B CA. Behaviour, con-dot-ta, f. mi-ni. * The pupil himself must examine whether he is to use before any noun or adjective the article or not, the prepositions di, a, and da, only being occasionally employed to denote the genitive, dative, placed in the order in which they are to be translated into Italian. I have thought it useful, in some cases, to denote the pronunciation of the zor zz. I have done so by placing after such words in parenthesis ts, thus (ts), when the pronunciation of the zor zz is to be the sharp, hissing one; and ds, thus (ds), when the pronunciation of the # or zz is to be the soft one. and ablative. It is, moreover, to be noted, that the words are For if the side a c be not greater than the side A B, it must be either equal to or less than the side A B. First, the side AC is not equal to the side A B; for, if so, the angle ABC is equal to the angle A CB, by Prop. V., which is contrary to the hypothesis; therefore, the side A c is not equal to the side a B. Second, the side A c is not less than the side AB; for, if so, the angle ABC is less than the angle AC B, by Prop. XVIII., which is also contrary to the hypothesis; therefore, the side the side AC is not equal to the side A B. Therefore, the side A c is not less than the side AB; and it has been proved that Wherefore, the greater angle AC is greater than the side AB. of every triangle, &c. former is greater than, equal to, or less than the angle opposite | sible, let the straight line a K 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 E, as just proved, and 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 A F, and AF than AG; also from the point A, only two equal straight lines can be drawn to the straight line BC, one upon each side of A D. AK is by hypothesis equal to a E, therefore AK is equal to A H, by Axiom I.; but it has been proved that a H, 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 a K 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 ine 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; First, to prove that the two sides Fig. 20, A Produce B A to the point D, and make A D equal to A c by Prop. III. Join D c. Because, in the triangle AD C, the side DA is, by construction, equal to the side A c; therefore the angle A DC is equal to the angle A CD. But the angle B CD, by Axiom IX., is greater than the angle ACD; therefore, the angle BCD is also greater than the angle ADC, or B D 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 BD is greater than the side B C. Again, in the triangle ADC, the side A D is equal to the side Ac, 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 BCD, was proved to be greater than its side в c; theregreater 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. Because the straight line A D is by hypothesis perpendicular to the straight line BC, therefore, by Def. 10, each of the angles A DE and ADH is a right angle; and they are, therefore, 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 angle ADH of the triangle ADE, is, by Prop. XVI., greater than its interior and remote angle AED, therefore, also the angle ADE is greater than the angle AED; wherefore, by 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 AF and AG are also greater than the straight line AD. Wherefore of all the straight lines A D, 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 APD, therefore also the angle A D F is greater than the angle AFD. 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 D E, much more, therefore, is the angle AEF greater than the angle AFE; wherefore, in the triangle A EF, by Prop. XVIII., the side A F 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 A F. Therefore, of the other straight lines, AE is less than AF, and A F than AG; that is, the straight line nearer the perpendicular is always less than the more remote. Lastly, from the straight line DC, cut off the part DH equal to DE, by Prop III., and join A H. Because in the two triangles ADE and ADH, the two sides AD 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 A E, 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 Euclid. This exercise was solved by T. Borock, Great Warley; Quintin Pringle, Glasgow; J. H. Eastwood, Middleton, and others. Scholium.-Dr. Simson, in his notes to his edition of Euclid, makes the following proper remarks:-"Proclus, in his commentary [on Euclid], relates, that the Epicureans derided Prop. XX. as being manifest even to asses, and needing no demonstration; and his answer is, that though the truth of it be manifest to our senses, yet it is science which must give 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 propositions be not demonstrated." It is true that this Prop. XX. is merely a corollary to the definition of a straight line 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 the Epicureans; but these philosophers were accustomed to laugh and mock at everything that did not just exactly square 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. B Scholium 2.-This proposition may be demonstrated by | XV.; therefore, the angle AEG is equal to the angle FEG, by another method, as follows:-In fig. Axiom I. But the side A E is equal to the side EF, by construcv, let B A C be a triangle, and let it be Fig. x. required to prove that the two sides BA and AC 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 DA C is greater than its interior and remote angle DA C, by Prop. XVI., and the exterior angle CDA of the triangle BDA is greater than its interior and remote angle D A B; and that the angles DAC and DA B are equal; therefore, the angle B DA is also greater than the angle DAB, 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 A c are greater than the whole side BC. B In any other case, let BA O, fig. w, be a triangle, of which the side B c is greater than the side B A; then the remaining side a c is greater than the difference between the other two sides, B C and B A. From BC, the greater side, cut off в D, a part equal to the less side B A, by Prop. III. Because the two sides B A 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 c D, 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. 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 A E, by Prop. III. Join FB, and let it cut CD in the point G. Join AG. 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 à ¤ and в н. Because the angle AEG is equal to the angle A EC by construction, and the angle & E C equal to the angle EG by Prop. D E H tion, and the side EG is common to the two triangles AEG 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 FB. But it has been shown that A G and G B are equal to FB; thefore A H and H в are together greater than AG and GB, that is, AG and GB are together less than the sum of a 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 o 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. Scholium 2. In the preceding demonstrations, it is very properly remarked by T. Bocock, Great Warley, that this axiom is taken for granted, viz. "If one magnitude be greater another, and if the same or equal magnitudes be added to each, 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."* * The exercises on Problem XX., were solved as follows: I., II, and III. by J. H, Eastwood, Middleton; E. J. Bremner, Carlisle; T. 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 Thanks.-W. LESSONS IN BOOKKEEPING.---No. X I. 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 197 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 preceding for once, therefore, we leave this subject in their hands. All ness for twenty, thirty, aye, and forty years, have thanked us we shall say, is this: that gentlemen who have been in busipersonally many times for the lessons on this subject which they have received from us, and particularly in reference to end of this Journal, but which cannot be fully explained in this our method of striking a General Balance, exemplified at the 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 London and Westminster Bank Sundries Dr. To Cash, as per C. B. fol. 1 London and Westminster Bank Petty Cash Account Three Per Cents. Consols Private Account Sundries Dr. ... ... ... To Bills Payable, as per B. P. B. fol. 1 Osmond and Co. Andrews and Co. Cotton Account Dr. To Sundries 4 To Osmond and Co. 5 ... To Andrews and Co. February. JOURNAL. : : : 10 985 10 CR. 0 £1200 40 0000 288 3 463 3 1005 (1) 288 463 £5912 14 (2) |