the Island of Ischia in the Bay of Naples, has been produced since the historical period. The l'ost Pleistocene formations about Naples show that very great changes have taken place throughout the whole of the volcanic district of Campania during the last two thousand years. One of the most remarkable is the formation of the modern cone of Vesuvius since a D. 79, which is represented in fig. 2. Before the year just mentioned, Mount Vesuvius might have been regarded as an extinct volcano, but at that period the rocks of the mountain were blown to pieces and fell into the gulf beneath, and its cliffs form the circular ridge, called Somma, which is several miles in diameter, the highest point of which appears to the left of the engraving. The enormous cone in the centre has been formed since the year A.d. 79. In the Bay of Baiæ, not far from Naples, there is an entire mountain, consisting of pumice and ashes, a mile and a half in circumference, and 450 feet high, which was formed by an earthquake on Sept. 29, 1538. During this catastrophe the north coast of this bay was permanently elevated twenty feet, exhibiting tufaceous strata filled with articles fabricated by man, such as fragments of sculpture and pieces of pottery, which are everywhere mingled with marine shells. Examples of raised beaches, of shell marl, and of submerged forests, are found in almost every part of the world; and they, in the majority of instances, afford proofs that they have been occasioned by physical changes of very modern date. These Post Pleistocene groups of rocks claim the particular attention of young geologists, as they furnish us with the clearest instances of the harmony between Geology and Revelation; for these rocks establish the fact stated by Revelation, that MAN is among the latest of the animals created to inhabit this earth. They show also that the epoch when the existing races of plants and animals were placed on the earth, must have been recent. Suppose that human remains, say bones or implements, had been found in ancient rocks which can be proved to have been formed at the bottom of deep oceans, or that they were found mingled with the earliest organic fossils in the Silurian rocks, it would have been impossible, for geologists, at least, to reconcile the two records of the Almighty Creator. Instead of this, geology proves, in harmony with Scripture, that the introduction of man among the creations of earth has not been of very remote antiquity. 1. THE LOESS OF THE RHINE. This Loess is sometimes called Lehm, and consists of a deposit of yellowish marl, often not less than fifty feet thick, abounding with calcareous concretions and sandy nodules. It is this kind of rock that forms groups of low hills at the foot of each mountain chain that enters the river valley. It is sometimes as high as 1,500 feet above the level of the sea, showing that the bed of the Rhine was once at that elevation. It forms also the subsoil of the plains on which Coblentz and Bonn are situated, and extends up as far as the falls of Schaffhausen, where it is seen to repose on beds of rolled flints and other pebbles of the drift period. It contains land shells and freshwater shells of many existing species; but the only mammalian remains found in it, are a few bones of the horse and the mammoth. 2. THE REGUR OF INDIA. The Regur is the cotton soil which covers one-third of all southern India, ranges northward to a great distance, and extends into the Burman empire. Its colour is bluish black, greenish or dark gray. Its thickness varies from three to twenty feet. 3. THE TILL. In almost every district of the globe there is found under the vegetable mould that covers the surface of the earth, a deposit of sand, mud and loose gravel, which has been called alluvium, from alluo to wash, and alluvio, an inundation. It is called by this name, because the bed of gravel has every appearance of having been spread by a flood, and the grains of sand and pebble appear as if they had been rolled by water, and had formed the bed of a mighty river. It is found in the higher latitudes of North America and of Europe, where it extends from Finland and the Scandinavian mountains to the North of Russia, and the low countries bordering on the Baltic, and on the eastern coast of Scotland and England. This deposit consists of sand, mud and clay, sometimes in a stratified state, but very often wholly unstratified, having a depth of more than a hundred feet. It is the unstratified part of this formation that is called by geologists "the Till." It contains numerous fragments of rocks, some angular, some rounded, derived from formations of all ages. Some of the blocks are of immense size. This rock is almost everywhere destitute of organic remains, except where they have been washed into it from older formations. 4. THE BOULDER FORMATION. When this formation contains large blocks of ancient rocks it is called "the drift," and "the boulder formation," whose probable connexion with floating ice has already been considered. Wherever it has been examined in Russia, it has been found throughout to be superimposed upon strata that contain recent shells, and that, consequently, the accumulation is post pleistocene. The same is the case about Upsala in Sweden. Everywhere it shows that the transport of erratic boulders continued to take place after the North of Europe had assumed its present physical features. It has been intimated already that it is next to impossible to mark the limits between chronological time and geological epochs. You have Been thut some beds in this Post Pleistocene group give indubitable proof that they were formed after the creation of man. Still, there are other beds, lying lower in this group, that present satisfactory evidence that they were deposited before man came upon the earth. First, no fossils of human bones, and no relies of human art, have ever been found in them. Secondly, in none of these rocks, formed as alluvial beds in the waters of the ocean, have any human remains of any kind been discovered. Thiroly, nevertheless the remains of Of the University of Pavia, and Professor of the German and Italian animals and of plants, identical or very similar to the existing races, are found in the lower formations of the group. Fourthly, a comparison of these beds with the physical conditions of the globe at the beginning of the Pleiocene period, shows that the state and aspect of the earth were very similar to the present, and that this similarity continued to increase till we approach the historical era. LESSONS IN ITALIAN GRAMMAR.-No. XVII. By CHARLES TAUSENAU, M.D., Languages at the Kensington Proprietary Grammar School. EXERCISES.-ITALIAN-ENGLISH. Ha man-dá-to la lêt-te-ra a Gio-ván-ni. Ti-rá-re ad un uccêl-lo. Il mer-cán-te pên-sa al gua-dá-gno. Tóc-ca un fiori-no ad ú-no. O-gnu-no tí-ra l'á-cqua al sú-o mo-lí-no. un Dál-le pa-rô-le si vên-ne ál-le ba-sto-na-te. A chi l'a-vé-te The districts of Italy, to which your attention has been already mo-strá-to? a Piê-tro o al-la cu-gi-na? A che pen-sú-te? directed, abound in proofs and illustrations of this statement. pên. so all' av-ve-ní-re. Ar-ri-ve-ré-mo prê-sto úl-la prôs-siin the Bay of Baia there are, besides the beds of tufa just men- ma pô-sta? E'-gli è cór-so sú-bi-to ál-la pôr-ta. Par-lá-va tioned, other tufaceous beds of a date evidently anterior to the ad ú-no stra-niê-ro. Lo in-ci-tò ál-la côl-le-ra. Pre-fe-rí-sce origin of man. These rocks are so thick as to form hills of from il bê-ne al má-le. La sú-a con-ver-sa-zió-ne mi viê-ne a nô-ja, 500 to 2,000 feet high. These beds contain all the marine shells E-gli se lo rê-ca a dis-o-nó-re. La li-be-ra-li-tà gli viên imnow abounding in the neighbouring sea, and yet they are inter-pu-tá-ta a di-fêt-to. E's-si ê-ra-no ál-la các-cia, ál-le nôz-ze, stratified with different sheets of lava. In the same manner, à prán-zo, a cé-na, al fe-stí-no. An-dré-te do-má-ni al ri-dótsimilar beds, consisting of clay and volcanic tufa, rise, in the to? al con-cêr-to? I'-o an-drò do-má-ni a un bál-lo. An-dáneighbourhood of Naples, to the height of 1,500 feet above the sea; te a im-pe-rá-re, a scrí-ve-re, a dor-mi-re, a man-giá-re. but these differ much from the kindred strata at Puzzuoli, for they si ván-uo a spás-so, a pas-seg-giá-re.. An-dia-mo al caf-fè. contain no relic or trace whatever of the existence of mau. Per dó-ve si va ál-la pô-sta? ál-la do-gá-na? E'-gli è a BerIn this lower division of the Post Pleistocene group are classed lí-no. Sog-giór-na in Fi-rên-ze. E-gli mo-rì in Not-tin-ga| the Loess of the Rhine, the Cotton soil of India, the Till of Eng-mo. E'-gli lo con-dur-rà a Cê-stria. El-la giún-se a Lió-ne. land and Scotland, and the newer part of the Northern Drift. E'-gli è ar-ri-vá-to in Bri-stôl. E'-gli è ná-to in Pli-mút-te. a E's 1 L'I-sti-tu-to po-li-tê-cni-co in Pa-ri-gi. La pô-sta pár-te ó-gni dì per l' I-ta-lia, per Ve-nê-zia, per Ró-ma. E'-gli dê-ve recar-si a Mi-la-no. E' re-sta-to tût-to il giór-no a ca-sa. E'-gli non va a pa-láz-zo, a cór-te. Di qui a Neo-ca-stêl-lo, a Jork. Ha mandato, he has sent. VOCABULARY. Tirare, to draw, trail, drag; to shoot or fire, &c. Uccello, bird. Tocca, falls to the lot or share mill (tirar l'acqua al suo molino, to convey water to one's mill; to look to the main chance or to number one). Da, from. Parola, word. Si venne, one (they) came. | ENGLISH-ITALIAN. * Thy mother has lost her umbrella. My sister has found a pen. Where have you bought this penknife? Hast thou seen our horse? We have seen a large inn. Your little brother has a good watch. Our brother is tall, but our sister is little. I have a hat which is very fine. The watch which you have bought is good. Our uncle has received a letter. This son has lost his mother. This daughter has lost her Nozze (ts), f. pl., wedding, mar- father. This present is for this child. riage feast. Erano, were. Caccia, chase. Pranzo, dinner. Cena, supper. ENGLISH-ITALIAN. Mr. Thomson has gone to the exchange. Let us go into the Festino (dancing, gaming, &c.), concert. The sisters have gone to-day to the evening enter evening party. Io andrò domani, I shall go to morrow. Caffe, coffee, coffee-house. Dogana, custom-house. Prossimo (m.), prossima (F.), | Ella giunse, she arrived. next. Lione, Lyons. Parte, starts. Ogni, every. | tainment. He is at the ball, and the brother in the concert. We have paid a visit to the neighbour; he lives on the second floor, and the son on the ground floor. We are now sitting at table. Think of more serious things. The misers are like the horses that carry wine and drink water, and like the asses that bear gold and eat thistles. He lives at the Black Eagle, and not at the Golden Lion. I have spoken to him at the coffee-house. Shall we play a game at cards or at chess? Mr. Thomson, Has gone, è an-dá-to VOCABULARY. Si-gnór ‡ | Are like, ras-so-mí-glia-no Let us go, an-điá-mo Evening entertainment, con- We have paid, ab-biá-mo fút-to Second floor, se-cón-do piá-no, m. Ground floor, pián ter-ré-no,m. Table, tá-vo-la, f. Think, pen-sá-te (i. e. direct Da Carry, mé-na-no Wine, ví-no, m. Black Eagle, á-cqui-la né-ra. f. Golden Lion, león đô-ro, in. (with the preposition a) I have already stated that the particle di denotes a mere mental separation of ideas or notions, while the particle da * After a careful study of the previous colloquial exercises, even Egli deve recarsi, he must de-ordinary pupils will be quite able to translate the following sentences without the aid of a vocabulary. part. Milano, Milan. E restato, he has remained. + In Italian, tall and great frequently are expressed by the same word. T When the word Si-gno-re is followed by a noun, the final e is Egli non va, he does not go. dropped, except when the noun that follows begins with the s imPalazzo(ts), palace; court; guild-pure; e. g. il Si-gnór An-tô-nio, Mr. Anthony; il Signor Fran-cé-sco, hall, townhall, council- Mr. Francis; il Si-gnór cón-te, count; il Signor ba-ró-ne, baron; il Si-gnór dot-tó-re, doctor; il Si-gnór con-si-glie-re, counsellor; Si1 gno-re Stê-fa-no, Mr. Stephen., house (andare a palazzo, to * In some of its meanings, this word denotes discreditable places of resort, and, to avoid ambiguity, it should only be used with precaution in the above-stated signification. || The verbs giuo-cá-re, to play (at cards or at any other game), and fa-re ú-na par-ti-ta, to play a game or make up a match (t cards or any other game), invariably require the preposition a; e. g. giuo-cá-re a un giub-có, ai dá-di (or a dá-dí), ál-le cár-te, á gli scác chi (or a scác-chi), a tre-sêt-te, all' óm-bre, ál-la pál-la, a pic-chét-to, &c., to play at a game, at dice, at cards, at chess, at tre-sept (an Italian game at cards), at omber, at tennis, at piquet, &c.; jac-cid-mo na par-ti-ta al bi-gliár-do, al whist, al cribbage, &c., let us have a game or make up a match at billiards, at whist, at cribbage, &c. expresses a real separation of objects. This is the fundamental | gí-re, to fly, escape, &c., admit of the preposition di before that signification of da, and, on this account, it must be pronounced place from which the going away or departure takes place, to be the very opposite or logical antagonist of the particle a. this apparent deviation from the general rule, without diffiThis latter word indicates any kind of tangible or mental and culty, will be explainca by ellipsis; i. e. by the omission of the imaginary approach or direction to or towards a person or thing, preposition da, with some other general noun; e. g. ve-ni-re, while da expresses any kind of tangible or mental and ima-par-ti-re di Ró-ma (i. e. dúl-la cit-tà di Ró-ma), to arrive, to ginary, but clear and real separation, removal, distance, or depart from (the city of) Rome; é-gli è di A-ber-dô-nia (i. e, direction from a person or thing, and the ideas of direction to or dál-la cit-tù di A-ber-dô-nia), he is a native of (the town of) towards, and of a direction from a person or thing, are, to some Aberdeen; u-sci-re, sor-ti-re di cá-sa, di cór-te, di pa-láz-zo, dí extent, the very poles or extremities of all relations in which | teá-tro, di chié-sa, to go or come from home, from court, from words and things stand to each other; e. g. in this sentence, guild-hall, from theatre, from church. pár-lo di lui, I speak of him, it is evident that there is no direction whatever to or towards, but rather a direction from a person. This direction is, nevertheless, not sufficiently clear and real enough to justify the use of da; while, in the sentence vên-go da lui, I come from him, a real removal, distance, or separation from the person, from which I come, is understood, which can only be expressed by the particle da. As a further illustration, in the phrase un mer-cán-te di Ve-ró-na, a merchant of Verona, the particle di is a mere sign or intimation to distinguish the merchant from the town in which he lives. and not of his absence from it; while in the sentence é-gli viê-ne da Ve-ró-na, the particle da denotes an actual removal from that place. This fundamental explanation of the particle du, however, is not sufficient to convey a complete notion of all its uses; every language, generally speaking, being far too complex a vehicle of human thought anywhere to admit of such a summary discussion of its more important branches. Now, and hereafter, I shall be therefore obliged to explain the various modifications and exceptions of this general rule. The particle da, also, is used, in order, by naming the birthtion. The birth-place thus becomes, as it were, the surname of place, to distinguish one person from others of the same appellathe individual. This employment of da certainly is quite conformable to its fundamental notion, for the birth-place is a part of the general idea of origin, descent, or extraction; e. g. Gio-ván-ni da Fiê-so-le, Piê-tro da Cor-tó-na, Leo-nárdo da Vin-ci, Gui-do da Siê-na, Po-li-dô-ro da Ca-ra-vág-gio, Ra-faêl-lo da Ur-bí-no,* &e. Da, also, may denote any origin or commencement referring to time, and then it means since; e. g. da che vi ví-di la prí-ma volta, since (that day when) I saw you the first time; dál-la mí-a gio-va-néz-za in si-no qué-sto tém-po, since my youth till this day; dall' in-no pas-sa-to in qua, since last year; da dû-e né-si in qua, two months since; ddl-la môr-te di mi-o pá-dre in qui, since the death of my father.† The phrases da mat-ti-na, da sé-ra, đa nôt-te, mean: in the morning, in the evening, in the night (by night, at night); e. g. ô-pe-ra da far da mat-ti-na, work to be done in the morning; non e-sce da cá-sa che da sé-ra, he only goes from home in the evening; tá-li cô-se non si fán-no da not-te, such things are not done by night. The ideas of removal, distance, separation, dependence, deduction, or derivation, and origin or descent, are, as it were, only collateral or subordinate branches of the fundamental notion of a direction from a person, or thing, and that word (person or thing), the Da also signifies about, nearly, close upon, not far off from, removal, distance, deduction or derivation, origin or descent &c., e. g. hó gua-da-gnú-to da cín-que li-re ster-li-ne, I have from which, and the dependence on which, is expressed, re-gained or won about five pounds sterling; ho per-dú-to da sêi quires the particle da before it; e. g. scó-sta-ti da qué-sto luo- a ôt-to tál-le-ri, I have lost from about six to eight dollars; da go, begone from this place; al-lon-ta-ná-re ú-nic da un luô-go, to Ró-ma a Ná-po-li sa-rán-no da cên-to ses-sán-ta mí-glia, it is remove one from a place; ca-vá-re á-cqua dal póz-zo, to draw about a hundred and sixty miles from Rome to Naples; é-gli water from the well; ve-ni-re da lon-tá-no, to come from | vi re-ste-rà da cín-que a séi giór-ni, he will stay there from about afar; í-o vên-go dal giar-dí-no, da cá-sa, I come from the five to six days; sti-má-va-si a-vé-re in Fi-rên-ze da no-van-tagarden, from home; l' uc-cêl-lo è u-scí-to dál-la gáb-bia, the bird | mí-la bóc-che tra uô-mi-ni e fém-mi-né e fan-ciúl-li, about ninety has flown out of the cage; ac-cat-tá-re pá-ne da ú-no, to beg thousand persons, men, women, and children, were estimated one's bread of one; ciò (pron. ciô) di-pên-de dúl-la for-tú-na, da to be in Florence. voi, that depends on good luck, on you; dé-dúr-re ú-na ra-gióne da un prin-cí-pio fál-so, to deduce an argument, proof, or sanctioned by a universal usage, for the most part in the A logical contradiction and anomaly, though introduced and evidence from a false principle; dál-la qual cò-sa ná-cque-ro di-place of the preposition a, the constant employment of ver-se pa- ú-re, from which from which arose various fears; de-ri da in connexion` with connexion with those verbs which, with some vá-re l' o-rí-gi-ne di ú-na cò-sa da un' al-tra, to deduce the origin house, mansion, apartments, lodging, or any other place of one thing from another; di-ví-de-re ú-na cô-sa da un' al-tra, of continuance, denote any kind of motion to or towards, to separate one thing from another. any kind of living or residing with, and any kind of visit It is obvious that the idea of origin, expressed by da, neces-paid to, a person; e. g. an-dá-re dal mê-di-co, dal cal-zo-la-jo, sarily includes any action proceeding from a person or place. For to go to the physician, to the shoemaker; do-má-ni ver-rò this reason, on the one hand, the English preposition by, when- da voi, I shall come to you to-morrow; i-o vi me-ne-rò da ever in connexion with passive verbs it denotes cause, author-lui, I shall conduct you to him; ve-ni-te da me, dal mership, instrumentality, &c., must be translated by da; and, on cán-te, come to me, to the merchant; só-no stá-to da lui, dal the other hand, all verbs expressing a going away, or depar-fra-tel-lo, I have been at his, at the brother's house (with ture, generally demand this particle; e. g. Cur-tu-gi-ne fu fub- him, with the brother); á-bi-ta, al-lôg-gia da sú-o zí-o, he lives bri-cá-ta da Ďi-dó-ne, Carthage was built by Dido; fu e-gli da | or resides with his uncle. al-cú-ni suô-i se-gré-ti ne-mí-ci ac-cu-sú-to, he was accused by some of his secret enemies; a qué-sto giar-dí-no l' ú-cqua è ab- Da is sometimes a substitute for di; e. g. li bra-si-má-va dubon-de-vol-mén-te som-mi-ni-strá-ta da ú-na fre-schis-si-ma fon-follia, di codardia), he severely blamed them, now for their fon-ra-men-te, ó-ra da fol-li-a, ó-râ da cô-dăr-di-ä (instead of di tá-na, the water for this garden * is abundantly supplied by a very cool fountain; é-gli è par-tî-to da Lôn-dra, he has de- folly, now for their cowardice; és-si hàn-no mól-ti mô-di da alparted from London; co-min-ciò a an-da-re da Na-za-rêt-te leg-gia-re o da pas-sa-re quél-lo (instead of di alleggiare, di a Ge-ru-sa-lêm-me, be began to go be began to go from Nazareth to passare), they have many means to make it easier or to pass Jerusalem. Whenever the verbs u-sei-re or sor-ti-re, to go or come out or from; par-ti-re, to set off, depart; ve-ni-re, to come; fug Dd, as well as the English by, is in these cases the preposition, which must be placed before the nominative case of the original sentence of the active voice whenever the latter is to be changed into the passive; e. g. ú-nä fre-schís-si-ma fön-tá-na som-mi-ní-stra ab-bon-de-vol-men-te ở aqua a quisto giar-đê-no, a very cool fountain abundantly supplies the water for this garden. over t. of ellipsis that I mentioned only serve the purpose of The particle da can never be really omitted, and the cases grammatical explanation. * The English learner will, perhaps, best understand this use of da by translating it with sprung from. + Since (denoting time, and not in the sense of as or because) is translated by fin đu, được ...in giữ, or đó-pô, when it precedes a fîn dú-po, a noun. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides. equal to them, of the other; the base of that which has the greater angle is greater than the base of the other. In fig. 24, let A B C and DEF be two triangles. which have the two sides A B and AC equal to the two sides DE and D F, each to each; viz., A B equal to D E, and AC to D F. But the B angle B A C greater than the angle EDF. The base BC is greater than the base E F. Of the two sides DE and DF, let DE be the side which is not greater than the other. At the point D, in the straight line D E, make (I. 23) the angle E D G equal to the angle BA C. Make DG equal (1. 3) to AC or D F. And join E G and G F. Because DE is equal (Hyp.) to AB, and DG (Const.) to AC, the A Fig. 24. A Fig. 3. D E E G F two sides, ED and DG, are equal to the two BA and AC, each to each; and the angle EDG is equal (Const.) to the angle B AC; therefore the base EG is equal (I. 4) to the base BC. Again, because DG is equal to DF, the angle D F G is equal (I. 5) to the angle DGF; but the angle DGF is greater (Ax. 9) than the angle EGF; therefore the angle DFG is also greater than EGF; much more then is the angle E F G greater than the angle EGF. Now, because the angle EFG of the triangle EFG is greater than its angle EGF, and the greater (I. 19) angle is subtended by the greater side; therefore the side EG is greater than the side EF But EG was proved to be equal to BC, Therefore B C is greater than E F. Therefore if two triangles, &c. Q. E. D. Fig. 4. કે D G G Next, if the point F falls within the triangle DE G, as in fig. 3., DF and F E, taken together, are less than DG and GE taken together (I. 24); but DG is equal to DF (Const.). Therefore E G is greater than EF (Axiom 5); but E G was shown to be equal to в C, therefore BC is greater than EF. The case in which the point r falls without the triangle D E G, as in fig. 4, is demonstrated in the same manner as above, in the proScholium 1.—Dr. Simson, in the construction of this proposi- position itself, and therefore it need not be repeated here. Wheretion, introduced these words: "of the two sides D E, DF, let DE before the exercise is demonstrated. Q. E. F.* the side which is not greater than the other," in order to avoid three distinct cases of construction, which would arise by taking that side which is greater than the other. Scholium II.-It has been remarked that Euclid's demonstration of this proposition appears to be defective, because of the omission of the words introduced by Dr. Simson, as stated in the preceding consideration of the three cases referred to Scholium. But upon in the following exercise, it would appear that Euclid had originally contemplated their insertion, inasmuch as the second case of it, as demonstrated below, requires only a simple and direct reference to Prop. XXI. Now Euclid is not guilty, in general, of bringing in propositions, in any book, which do not bear upon those that folfow; but it has been universally admitted that Prop. XXI. was of this description; now if he considered the three cases of Prop. XXIV., this objection is at once removed. Why they are not found in the common Greek text, we cannot tell at present. EXERCISE I. TO PROPOSITION XXIV. EXERCISE II. TO PROPOSITION XXIV. Demonstrate that, in Dr. Simson's construction, the straight line EG cuts the straight line DF in some point between D and F. point H; it is required to demonstrate In fig. z. let DF meet EG in the that the point H lies between the D. points D and F. Because D E is less than D G (Hyp.), the angle DGE is less (I. 18) than the greater (1. 16) than the angle D E G ; fol-angle DFG. But the angle D H G is greater (1. 16) than the angle DEG; much more, then, is the angle D H & fore the side DG is greater than the greater than the angle D G B. Thereside DH (I. 19). But D G is equal to D F (Const.). Therefore DF is also greater than D H. Therefore EG cuts D F in the point н, between the points D and F. Q. E. D.* Demonstrate this propositio, by making the construction on the greater of the two sides of the triangle DEF, and exhibit the three istinct cases above mentioned. Let ABC, fig. 1, and DEF, figs. 2, 3, and 4, be two triangles which have two sides of the one equal to two sides of the other, each to each, viz., the side A B to the side D E, and the side A c to the side D F, but the angle B A C greater than the angle ED F. The base C is greater than the base ET. than the other. At the point D in the straight line D E make the Of the two sides DE and DF, let DE be that which is greater angle EDG equal to the angle BAC (I. 23). Make D G equal to A C or D F (I. 3) and join E G. Because DE is equal to A B (Hyp.) and DG to A c (Const.), and the angle EDG to the angle BA c (Const.); therefore the base EG is equal to the base B C (I. 4). Now, if the point F falls upon E G, EG, Fig. z. If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one greater than the base of the other; the angle contained by the two sides of that which has the greater base, is greater than the angle contained by the two sides equal to them of the other. sides A B and A C equal to the two sides D E and 1> F, each to each; In fig. 25, let A B C and D E F be two triangles which have the two viz., A B equal to D E, and AC to D F ; but the base B C greater These exercises were solved by J. H. Eastwood, Middleton; C. L. Hadfield, Bolton-le-Moors; and Q. Pringle, Glasgow. than the base E F. The angle ED F. angle B A C is greater than the is equal (Const.) to the side EF, and the side a в to (Hyp.) Fig. 25. A D For, if the angle BAC be not greater than the angle ED F, it must either be equal to, or less than the angle ED F. The angle B A C is not equal to the angle EDF, because then the base BC would he equal (I. 4) to the base EF: but it is (Hyp) not equal. Therefore the angle B A C is not equal to the angle EDF. Again, the angle BAC is not less than the angle EDF, because then the base B C would be less (I. 24) than the base EF: but it is (Hyp.) not less. Therefore the angle B A C is not less than the angle EDF. And it was shown that the angle B A C is not equal to the angle EDF. Therefore the angle B A C is greater than the angle EDF. Wherefore, if two triangles, &c. Q. E. D. B B F Corollary. If two triangles have two sides of the one respectively equal to two sides of the other, the base of the one is greater than, equal to, or less than the base of the other, according as the angle opposite to the base of the one is greater than, equal to, or less than the angle opposite to the base of the other. PROPOSITION XXVI.--THEOREM. If two triangles have two angles of the one equal to two angles of the other, each to each; and onc side equal to one side,-viz., cither the sides adjacent to the equal angles, or the sides opposite to equal angles in each; then their other sides are equal, each to each, and also the third angle of the one to the third angle of the other. In fig. 26, let A B C and D E F be two triangles which have the two angles A B C and B C A of the one, equal to the two angles DEF and EFD of the other, each to each; viz., A B C to D E F, and B C A to EFD. Also, let a side of the one triangle be equal to a side of the other. Fig. 26. No. 1. D the side DE; the two sides AB and BH are equal to the two (I. 4) to the base D F, and the third angle B A C to the third angle Scholium. The enunciation of this proposition may be thus simplified: If two triangles have two angles of the one, equal to two angles of the other, each to each, and a side of the one equal to a side of the other similarly situated as to the equal angles, the two triangles are equal in every respect. The demonstration might also be conducted on the principle of supraposition, employed in the 4th and 8th propositions of this Book. This will form a good exercise for our students, and we leave it for them accordingly. EXERCISE I. TO PROPOSITION XXVI. In an isosceles triangle, if a straight line be drawn from the angle opposite the base, bisecting the angle, it bisects the base; or, if it bisect the base, it bisects the angle; and in cither case, it cuts the base at right angles. Fig. a. C D B In fig. a, let A B C be an isosceles triangle, of which the sides a C and CB are equal; and first, let the straight line CD bisect the angle А СВ. Then the base A B is bisected at D. Because, in the two triangles ACD and BCD, the two sides a C and CD are equal (Hyp.) to the two sides BC and CD, and the angle ACD is equal (Hyp.) to the angle BCD; therefore the base a D (I. 4) is equal to the base D B. A B is bisected in D. Also, by I. 4, the remaining angles of the triangle A CD are equal to the remaining angles of the triangle BCD, each to each, viz., those to which the equal sides are opposite; therefore, the angle ADC is equal to the angle BDC; but these are adjacent angles; therefore (Def. X.) they are right Fangles. No. 1. First, let those sides be equal which are adjacent to the angles that are equal in the two triangles; viz., B C equal to EF. Then their other sides are equal, each to each; viz., AB to DE, and AC to DF; and the third angle BAC of the one is equal to the third angle EDF of the other. For, if AB be not equal to D E, one of them must be greater than the other. Let A B be the greater of the two. Make B G equal (I. 3) to D E and join & c. A B Wherefore Secondly, let the straight line CD bisect the base A B. Then the angle A C B is bisected by c D. Because, in the two triangles ACD and B CD, the two sides A C and CD are equal (Hyp.) to the two sides B C and CD, and the base A D is equal (Hyp) to the base DB; therefore the angle a C D is equal (I. 8) to the angle B C D. Wherefore the angle ACB is bisected by CD. In the same manner, it may be shown that the angle ADC is equal to the angle BDC; but these are adjacent angles; therefore, by Def. | X. they are right angles. Because, in the two triangles G B C and D E F, the side B G is equal (Const.) to the side D E, and the side в C (Hyp.) to the side E F, the two sides G B and B C are equal to the two sides D E and E F, each to each. But the angle G B C is equal (Hyp.) to the angle D E F. Therefore the base & c is equal (I. 4) to the base DF, and the triangle G B C to the triangle D E F; and the remaining angles of the one are equal to the remaining angles of the other, each to each; viz., those to which the equal sides are opposite. Therefore the Q. E. D.* | angle GCB is equal to the angle DFE. But the angle D F F is (Hyp.) equal to the angle B CA. Wherefore also the angle RCG is equal (Ax. 1) to the angle B C A, the less to the greater, which is impossible. Therefore the side A B is not unequal to the side DE; that is, A B is equal to DE; also B C is equal (Hyp.) to EF. Therefore the two sides A B and B c are equal to the two sides D E and EF, each to each; and the angle A B C is equal (Hyp.) to the angle DEF. Therefore the base a C is equal (I. 4) to the base D F, and the third angle B A C to the third angle ED F. No. 2. Next, let those sides which are Fig. 26. opposite to equal angles in each triangle a be equal to one another; viz., A B equal to DE. Then their other sides are equal; viz., A C to D F, and BC to EF: And the third angle B A C of the one is equal to the third angle E D F of the other. D HC E No. 2. EXERCISE II. TO PROPOSITION XXVI. Through a given point to draw a straight line which shall make equal angles with two straight lincs given in position. In fig. 6, let o be the given point and A B and CD the two straight lines given in position. It is required to draw through o, a straight line which shall make equal angles with A B and C D. Produce AR and CD till they meet in E; bisect the angle A E C (I. 9) by the straight line E F. From the point o (I. 12) draw OP at right angles to EF, and produce it to Q, so as to meet A B and CD in the points ® and s. Then o Q is the straight line required. Because in the two triangles REP a SEP, the angle REP is equal (Const.) to the angle S EP, and the angle RPE to the angle 8 PE, each of them being a right angle, for RP E is equal (I. 15) to SPF; and the side EP is common to both; therefore (I. 26) their other angles are equal, viz. the angle PRE to the angle PSE. FWherefore a straight line o Q has been drawn through the point o, For, if в c be not equal to EF, one B of them must be greater than the other. Let B c be the greater of the two. Make BH equal (I. 3) to E F, and join a H. This exercise should have been been appended to Prop. VIII.; it was solved by E. L. JONES (Pembroke Dock); Q. PRINGLE (Glasgow); Because in the two triangles A B H ́and DEF, the side B HD. H. DRIFFIELD; E. JONES; E. J. BREMNER (Carlisle); and others. |