PUNCTUATION is peculiar to the modern languages of Europe. It was wholly unknown to the Greeks and Romans; and the languages of the East, although they have certain marks or signs to indicate tones, have no regular system of punctuation. The Romans and the Greeks also, it is true, had certain points, which, like those of the languages of the East, were confined to the delivery and prounciation of words; but the pauses were indicated by breaking up the written matter into lines or paragraphs, not by marks resembling those in the modern system of punctuation. Hence, in the responses of the ancient oracles, which were generally written down by the priests and delivered to the inquirers, the ambiguity-doubtless intentional -which the want of punctuation caused, saved the credit of the oracle, whether the expected event was favourable or unfavourable. As an instance of this kind, may be cited that remarkable response which was given on a well-known occasion when the oracle was consulted with regard to the success of a certain military expedition, Ibis et redihis nunquam peribis in bello." Written, as it was, without being pointed, it might be translated either "Thou shalt go, and shalt never return, thou shalt perish in battle," or "Thou shalt go and shalt return, thou shalt never perish in battle." The correct translation depends on the placing of a comma after the word nunquam, or after redibis. The invention of the modern system of punctuation has been attributed to the Alexandrian grammarian Aristophanes, after whom it was improved by succeeding grammarians; but it was so entirely lost in the time of Charlemagne, that he found it necessary to have it restored by Warnefried and Alcuin. It consisted at first of only one point, used in three ways, and sometimes of a stroke, formed in several ways. But as no particular rules were followed in the use of these signs, punctuation was exceedingly uncertain, until the end of the fifteenth century, when the learned Venetian printers, the Manutii, increased the number of the signs, and established some fixed rules for their application. These were so generally adopted, that we may consider them as the inventors of the present method of punctuation; and although modern grammarians have introduced some improvements, nothing but a few particular rules have been added since their time. The design of the system referred to was purely grammatical, and had no further reference to enunciation, than to remove ambiguity in the meaning and to give precision to the sentence. This, therefore, is the object of punctuation, and although the marks employed in written language may sometimes denote the different pauses and tones of voice which the sense and accurate pronunciation require, yet they are more generally designed to mark the grammatical divisions of a sentence, and to show the dependence and relation of words and members which are separated by the intervening clauses. The teacher, therefore, who directs his pupils to "mind their pauses in reading," gives but an unintelligible direction to those who are unversed in the rules of analysis. A better direction would be to disregard the pauses, and endeavour to read the sentence with just such pauses and tones as they would employ if the sentence were their own, and they were uttering it in common conversation. Indeed it is often the case that correct and tasteful reading requires pauses, and these too of a considerable length, to be made, where such pauses are indicated in written language by no mark whatever. It is not unfrequently the case that the sense will allow no pause whatever to be made in cases where, if the marks alone were observed, it would seem that a pause of considerable length is required. The pupil, therefore, who has been taught to mind his pauses, must first be taught to unlearn this direction and endeavour to understand the sentence which he is to read, before he attempts to enunciate it. * The Hyphen, The Breve, The Apostrophe, The Brace, The Acute Accent, The Grave Accent, The Circumflex Accent, The Caret, The Cedilla, The Asterisk, The Section, The Paragraph, The Parallels, Ellipsis, sometimes expressed by Periods, thus, sometimes by Hyphens, thus, sometimes by Asterisks or Stars, thus, These characters, when judiciously employed, fix the meaning and give precision to the signification of sentences, which, in a written form, would be ambiguous or indefinite without them. Thus, "I said that he is dishonest it is true and I am sorry for it." Now the meaning of this sentence can be ascertained only by a correct punctuation. If it be punctuated as follows: "I said that he is dishonest, it is true, and I am sorry for it;" the meaning will be, that it is true that I said he was dishonest, and I am sorry that I said so. But if it be punctuated thus, "I said that he was Written here, of course, includes printed language. dishonest; it is true; and I am sorry for it" the meaning will be, I said that he was dishonest; it is true that he was dishonest, and I am sorry that he was so. Again, the following sentence, as here punctuated, is an innocent remark: "Believing Richard Brothers to be a prophet sent by God, I have painted his portrait." But the sentence, as it was originally written by its author, with the comma after sent, instead of after God, is a piece of horrid profanity. sound of the vowel, is called the Macron, from the Greek, signifying long. The mark called a Breve, indicating the short sound of the vowel, is from the Latin, signifying short. The word Ellipsis, also from the Greek, means an omission, and properly refers to the words, the members, or the sentences which are omitted, and not to the marks which indicate the omis sion. The word Apostrophe, also from the Greek, signifies the turning away, or the omission of one letter or more. The word apostrophe, as here used, must not be confounded with the same word as the name of a rhetorical figure. The word Diaresis is also from the Greek, and signifies the taking apart, or the separation of the vowels, which would otherThe term Accent is derived from the Latin language, and im. plies the tone of the voice with which a word or syllable is to be pronounced. A further instance of the importance of correct punctuation was afforded by a late advertisement, in which the commissioner for lighting one of the most commercial cities of Europe, by the misplacing of a comma in his advertisement, would have contracted for the supply of but half the required light. The advertisement represented the lamps as "4,050 in number, having two spouts each, composed of not less than twenty threads of cotton.' This ex-wise be pronounced as one syllable. pression implied that the lamps had each two spouts, and that the two spouts had twenty threads, that is, each spout had ten threads. But the meaning that the commissioner intended to convey was, that each spout had twenty threads; and his advertisement should have had the comma after "spouts," instead of after "each," thus: The lamps have two spouts, each composed of twenty threads, &c. These instances might suffice to illustrate the nature and the propriety of correct punctuation; but the following instance, known to many, will show the importance of the subject. The clerk of a congregation, in Scotland, had a paper handed to him, as the custom is, to read just before the minister stood up to pray The word Paragraph is derived from the Greek language, and with and for the congregation, containing the following words, un-signifies a writing in the margin. This mark, like that of the secpointed: "A man going to sea his wife desires the prayers of the tion, was formerly used to designate those divisions of a section congregation." The clerk read it as if a comma had been put at the which are now indicated by unfinished lines or blank spaces. end of the word wife, and unfortunately excited, in no small de- This mark, as well as the section, is now rarely used except as gree, the risible faculties of the people assembled :-thus, "A a reference. man going to sea (see) his wife, desires the prayers of the congregation." But although the meaning of a sentence is thus materially affected by the punctuation, it will be seen in the following lessous that the punctuation alone is an unsafe guide to follow in the enunciation of any collection of words. For, in many cases, these marks indicate no pause, emphasis, or other remarkable circumstance requiring notice in the enunciation of the sentence. The word Section, derived also from the Latin, signifies a cutting, or a division. The character which denotes a section seems to be composed of ss, and to be an abbreviation of the words signum sectionis, or the sign of a section. This character, which was formerly used as the sign of the division of a discourse, is now rarely used except as a reference to a note at the bottom of the page. It may further be remarked, that notes at the bottom of the page, on the margin, or at the end of the book, are often indicated by figures, or by letters, instead of the marks which have already been enumerated. The word Caret is from the Latin, and signifies it is wanting. This mark is used only in manuscripts. The Cedilla is a mark placed under the letters c and g to indicate the soft sound of those letters. The Asterisk, Obelisk, Double Obelisk, and Parallels, with the section and paragraph, are merely arbitrary marks to call attention to the notes at the bottom of the page. As these marks which have now been enumerated all have a The nature of the marks used in written language may also be understood by a reference to the origin of their names. The word Comma is derived from the Greek language, and properly designates a section, or part struck off from a complete sentence. In its usual acceptation, it signifies the point which marks meaning, and are employed for some special purpose, it is recomthe smaller portions of a period. It therefore represents the short-mended to the student never to pass by them without being assured est pause, and consequently marks the least constructive, or most that he understands what that purpose is. Correct and tasteful dependent parts of a sentence. reading can never be attained without a full appreciation of the The word Colon is from the Greek, and signifies a member of a meaning which the author intended to convey; and that meaning sentence, and the Latin prefix semi means half. Hence, a Semi-is often to be ascertained by the arbitrary marks employed by him colon is used for the purpose of pointing out those parts of a com- for the purpose of giving definiteness to an expression. At the pound sentence, which although they each constitute a distinct same time the student should consider these marks as his guide to proposition, have yet a dependence upon each other, or on some the meaning only, not to the enunciation of a sentence. Correct common clause. The Colon is used to divide a sentence into two delivery must be left to the guidance of taste and judgment otheror more parts, which, although the sense be complete in each, a. wise acquired. not independent. The word Period is derived from the Greek, and means a cius. or well-rounded sentence. Hence, when the circuit of the s completed, with all its relations, the mark bearing this na used to denote this completion. The word Interrogation is derived from the Latin, and means a question. Hence, the mark so called is put at the end of a question. The word Exclamation is from the same language, and means a passionate utterance. Hence, the mark so called is put at the end of such utterances. The word Parenthesis is derived from the Greek language, and means an insertion. A sentence, clause, or phrase, inserted between the parts of another sentence for the purpose of explanation, or of calling particular attention, is properly called a parenthesis. In many excellent selections for lessons in reading, the pieces have been arranged in regular order, according to the nature of their respective subjects, under the heads of Narrative, Descriptive, Didactic, Argumentative, and Pathetic pieces, Public Speeches, Promiscuous pieces, the Eloquence of the Bar, of the Pulpit and of the Forum. By Narrative pieces is meant those pieces only which contain a simple narration or story. Descriptive pieces are those in which something is described, chiefly from nature. Didactic pieces are those designed to convey some particular kind of instruction, whether moral, religious or scientific. Argumentative pieces are those in which some truth is designed to be proved in an agreeable manner. Pathetic pieces are those by which the feelings of pity, love, admiration and other passions, are excited. Promiscuous pieces are those which do not fall exclusively under any of the classes which have been enumerated, or which consist of a mixture of those classes. The Eloquence of the Bar consists of speeches (or pleas as they are technically called) made by distinThe word Hyphen is derived from the Greek language, and sig-guished lawyers in the courts of justice in favour of or against a nifies under one, that is, together; and is used to imply that the letters or syllables between which it is placed are to be taken together as one word. It is to be remarked, however, that the name parenthesis belongs only to the sentence inserted between brackets or crotchets, and not to those marks themselves. supposed criminal. The eloquence of the Pulpit consists of sermons or discourses delivered on religious occasions. The Eloquence of the Forum consists in the speeches, addresses, orations, The hyphen, when placed over a vowel, to indicate the long | &c., addressed to political or promiscuous assemblies. Whom he carries along with | Send me, man-di-te-mi Dozen, doz-zi-na (ds), f. A Lemon, li-mó-ne, m. Who follows him, che gli va Two, du-e Here are, éc-co Yard, brác-cio, m. (pl. le brác- Taffeta, taf-fe-tà, in. che a-vé-te do-man-dá-ta A Pound, lib-bra, f. che vi è stata man-da-ta To many, this information may seem superfluous or puerile. But ast hese lessons are designed for the young and the unlearned, it must not be forgotten that their sources of information are few and that they will not always take the pains to inform themselves of the meaning of words, even when they are familiar to their eyes in capital letters, and in the running titles of the books before them every day. It is often the case that the teacher also, taking for granted that his pupils are familiar with the meaning of words so often presented to their eyes, neglects to question them on the subject; and in riper years it becomes a matter of surprise to the pupil himself that, in early life, words which he had heard sounded almost every day at school presented no idea to his mind beyond that of an unmeaning, or rather an unintelligible sound. The object of all education is not so much to fill the mind with knowledge as to strengthen its powers, and enlarge its capacity. Those exercises, therefore, are always most beneficial in education, which tend most effectually to produce this result: There is, perhaps, no branch of study connected with popular education, which, when properly pursued, is more highly subservient to this end than the The use of this particle frequently coincides with the use of study of correct and tasteful reading, as an art. It necessarily the preposition to in English grammar. Generally speaking, involves a complete knowledge of the subject to be read, the rela- any kind of direction, expressed by a verb, to or towards a pertion and dependencies of the phrases, clauses and members of the son or thing, is denoted by this word. The ideas of similarity sentences, the proper meaning of the words employed, and the or resemblance, of approaching or approximation, of a direction or connexion between the sentences themselves. This cannot be mere reference to any thing, end, aim, or point of time, form, acquired without a vigorous employment of the perceptive powers, as it were, only parts or branches of this fundamental significaaided by those of comparison, of analysis, of reasoning, of judg- tion of the particle a, and whenever the action of the subject of ment, of taste, and of discrimination. Subordinate and auxiliary a sentence (i. e. of the nominative) expresses such direction or to the acquisition of this important art, the student is recommended approach to or towards persons or things, a must be placed to exercise also the power of classification, while studying a read- | before them; e. g. ac-cô-sta-ti ál-la tá-vo-la, approach thyself ing lesson (which should always be studied previous to practising to the table; al-cá-ne dá-te gli ós-si, give the bones to the dog; it), to ascertain under which of the above mentioned classes, il fí-glio ras-so-mí-glia al pá-dre, the son is like the father; ne whether narrative, descriptive, didactic, &c., the piece he is about par-le-rò al cu-gi-no, I shall speak of it to the cousin; al cánto read belongs. The student who thus employs his faculties cannot to si ri-co-nó-sce l' uc-cêl-lo, by the song one knows the bird; fail to feel a vigorous growth of intellect springing up in his owna-vi-ro non pên-sa che al da-ná-ro, the avaricious man only mind, and will be amply compensated for his labour, by a command over the stores of literature not to be gained by any other method. LESSONS IN ITALIAN GRAMMAR.-No. XVI. Of the University of Pavia, and Professor of the Italian and German Time, têm-po, m. VOCABULARY. Present, a-des-so (adv. now) Best, mi-glió-re He had hidden himself, égli si ê-ra na-scó-sto Room, stán-za (or cd-me-ra) Bridge, pón- te, m. Yours, vo-stro, m., rô-stra, f. Has only, ha so-la-men-te One, ú-no, m., ú-na, f. Wood, lé-gno, m. Edward, E-du-ár-do Has received, ha ri-ce-vu-to Watch, o-ro-lô-gin (o-ra-u-lo) Pair, pá-jo, m. Shoe-buckle, fib-bia, f. Steel, ac-cid-jo, 8. m. Once, ú-na vôl-ta They wore, si por-tá-va-no Dress, d-bi-to, m. Use, ú-80, m. Vessel, vá-so, m. Copper, ra-me, m. thinks of money; í-o lo dis-si al a-mi-co vô-stro, I told it your friend; é-gli lo diê-de a' pô-ve-ri, he gave it to the poor; -o vá-do a Ró-ma, I go to Rome; non cre-dé-te a ló-ro, do not believe them; dís-si a lui, an dá-te a cá-sa, I told him, go home (i.e. to the house); pic-chiá-re ál-la pôr-la, to knock at the door; scri-ve-re a qual-che-du-no, to write to somebody; ag. giu-ghe-re i-na cô sa ad un al-tra, to add one thing to another; ce-de-re sú-o-di-rit-to a qual-che-du-no, to transmit or cede one's right to any one; co-strin-ge-re ú-no ad ú-na a-zió-ne, to compel or force any one to some action; ver-rò a mêz-zo giór-no, a mézza nôt-te, il-le dú-e, al têm-po fis-sá-to, al pri-mo del mé-se, I shall come at noon, at midnight, at two o'clock, at the appointed time, on the first of the month. Phrases, not literally or strictly expressing an abode, residence, stay, continuance, or being in a place, but merely Has been prohibited, è stá-to nearness or presence, require the particle a and not in, which pro-i-bi-to Sweden, Svê-zia Shambles, bec-che-ri-a, (slaughter-house) Are for sale, si trô-va Meat, cúr-ne, f. Young ox, min-20, m. always denotes a real and not merely imaginary continuance or being in (i. e. in the interior of) a place or thing, cr some f.action taking place in it; e. g. é-gli è al bál-lo, he is at the ball; je-sti-no, at the (dancing and gaming) evening party; a táda on-la, at table; al con-cer-to, in the concert; a giuo-cd-re, at play or game; a stu-did-re, (engaged) in study. Wether, ca-stró-ne, m. (eat It is, és-so è Good, buô-no, m., buôm na, Ẵ Whom you see, che ve-dé-te From what has been explained, it is obvious that in those phrases which merely denote the moving, approaching, or tendency to or towards a place or thing, and not strictly the entering or penetrating into it, a and not in must be used; for in means the actual motion or penetration into the interior of any locality; e. g. i-o vá-do al bál-lo, I go to the ball; a távola, to table; a cé-na, to supper; a im-pa-rá-re, to learn, i, e. to (the pursuit of) learning; a giuo-cd-re, to play, i. e. to (the diversion of) playing. The proper nouns of towns, cities, boroughs, or similar localities, are an exception to the last-mentioned rule, for it is quite allowable indiscriminately to place a or in before them whenever the abode, residence, stay, arrival, continuance, or being in or within them (i. e. in their interior) is to be designated; e. g. é-gli è a or in Nú-po-li, he is at or in Naples; tro-van-do-si e-gli u-na vôl-ta a Pa-ri-gi, being once in Paris; él-la è ar-ri-vá-ta a or in Var-sá-via, she is arrived in Warsaw. The verbs par-ti-re, to depart, set out or off, and con-ti-nud-re, to continue, proceed on (one's journey), are another exception, for they require the preposition per before the name of that locality or even country, towards which a journey or any motion is directed; e. g. é-gli è par-tí-to per Co-stan-ti-no- | for life to the galleys; és-se-re sen-si-bi-le a qual-che co-sa, to feel po-li, per Pie-tro-bur-go, per la Sviz-ze-ra, he has started for compassion for (or to be susceptible of) something: by the Constantinople, for St. Petersburgh, for Switzerland; con-ti- preposition by; e. g. lo fa-rui a for-za, thou wilt do it by connu-a-re il sú-o viág-gio per la Po-bo-nia, per Mó-sca, to proceed on straint: by the preposition of; e. g. chie-de-re ad al-cú-nó, t one's journey to Poland, to Moscow. desire or require of somebody: by the word as; e. g. mêt-terNext to di, the particle a is of the most extensive use, and servant; a-vé-re a si-gnó-re, to have as a master: by at a time; si a sèr-vo con al-cu-no, to engage one's self to somebody as a though the relations in which this word stands to others are not quite so loose and vague as those of di, they are various. B. a dú-e a dú-e, two at a time, two and two: by adverbial enough to admit of modes of application which, even in expressions or phrases; e.g. a buon mer-cá-to, at a small price, Italian, might sometimes be more suitably dispensed with by gio, as bad as possible; dl-la rin-fu-sa, confusedly, promiscucheap; al-la sca-pe-strd-ta, licentiously, dissolutely; dl-la pêgthe use of prepositions of a more logical distinctness, and conse-ously; a mén-te, a me-mo-ria, by heart (to learn or know); a quently a greater clearness in special instances; e. g. mon-tá- bio-ca, by word of mouth; ve-ni-re al-le má-ni, to come to blows re a ca-val-lo (for só-pra un ca-vál-lo), to get or mount on horse- or to engage in close fight; an-dú-re a spás-so, a di-pôr-to, to, back; i-vi a pô-chi gior-ni ri-tor-nò (for dó-po pô-chi giór-ni), he take a walk; a quat-tro oc-chi, a té-sta a té-sta, in private, alone returned a few days after; fá-re a vo-lon-tà di cia-scu-no (for together (i. e. between four eyes, tête-à-tête); a ba-stán-za se-con-do la vo-lon-tu), to act according to, or to conform to the enough; a ma-no, at hand, near at hand, in readiness; with will of everybody; bat-té-an-si a pál-me (for cól-le pál-me), they fought with the palms of their hands; le rot-tu-re fu-ro-70 muor by the hand; artificially; by election; underhand, by fraud rá-te a pie-tra e a cal-ci-na (for con pié-tra e con cal-ci-na), the or deceit. breaches were walled up with stone and lime; non ci con-verrà com-bát-te-re a si pó-ca gên-te (for con-tra si pô-ca gén-te), it will not become us to fight against so few; mól-ti fan-no bé-ne a spe-rún-za di gua-dá-gno (for per i-spe-rán-za), many are honest through the hope of profit, &c. I have already stated that to avoid hiatus by a succession of vowels, generally ad, in the place of a, is used before a vowel, and I shall conclude this explanation of the uses of a by the remark that, in Italian classics, not a few passages, where at first sight the particle a appears to be a somewhat arbitrary of construction, will admit of a perfect elucidation by ellipsis. substitute for other prepositions or words, without any change Other uses, and some omissions of the particle a, will be commented on hereafter. If LESSONS IN GEOMETRY.-NO. XXV. (Continued from p. 196.) BOOK I.-PROPOSITION XXI.-THEOREM. from the ends of one side of a triangle, there be drawn twc straight lines to a point within the triangle, these together shalı be less than the other two sides of the triangle, but shall contain a greater angle. c, the ends of the side B c, let the two straight lines BD and In fig. 21, let A B C be a triangle, and from the points в and DC together shall be less than the other two sides B A and A C CD be drawn to the point D within the triangle. Then B D and of the triangle ABC, but shall contain an angle BDC greater than the angle BA C. Fig. 21. It is obvious that this variety of the significations of a will, for the purpose of translating it into English, require the use of many prepositions or other words, and sometimes even of adverbial expressions or phrases, which only practice and a patient method of reading good writers, by accurately compar ing the idioms and genius of the two languages, fully can teach. In a course of merely elementary lessons, I must naturally restrict myself to some, I think, useful hints in the following illustrations :-The particle a may be translated by the objective case (without any preposition); e. g. fé-re ve-de-re and al-củ-no ủng củ-sa, to let any one see something; do-man-dá-re ad al-cu-no, to ask one; toc-cá-re ad al-cú-no, to concern one; so-prav-vi-ve-re ad al-cú-no, to survive one; sup-pli-re a qual-che cô-sa, to complete or make up something: by the preposition to; e. g. ap-pli-car-si ad ú-na có-sa, to apply one's self to something; vôl-ger-si ad al-cuno, to turn to somebody; a si-ni-stra, a mán-ca, to the left; a dé-stra, to the right; an-dá-re, ve-ni-re a un luô-go, to go, come to a place; do-len-te a môr-te, grieved to death; passá-re a fil di spá-da, to be put to the sword (i. e. to the edge of the sword) by the preposition at; e. g. al le-vár del só-le, at sun-rise; al primo cen-no, at the first hint or sign; a mi-o senno, to my mind, liking, taste, fancy, will; se-dé-re a tá-vo-la, to sit at table; és-se-re (stá-re, tro-vár-sí) a un luô-go, to be at a place: by the preposition on or upon; e. g. a pé-na di môr-te, upon (or under) pain of death; af-fi-dár-si ad al-cú-no, to reckon Produce BD to E. Then, the two sides BA and A E, of the or build upon one; ap-po-giár-si a qual-che co-sa, to lean, rest, triangle ABE are greater than the third or to rely on something; in-si-ste-re a qual-che cô-sa, to insist side BE (I. 20). To each of these unon something; a piê-di, a ca-vál·lo, on foot, on horseback; a equals add EC, and the two sides BA and con-di-zió-ne, on condition; ad im-prê-sti-to, on trust or credit: A c are greater (A. 4) than BE and EC. by the preposition in; e. g. a dú-e mé-si, in two months; dl-la Again, the two sides CE and ED of the sfug-gi-ta, in passing by or in flight; di-pin-ge-re a 6-glio, to triangle CED are greater (I. 20) than the paint in oil; ve-sti-to a bian-co, dressed in white; dl-la fran-ce-third side CD. To each of these unequals, se, all' in gle-se, in the French, English manner or fashion; di- add DB, and the two sides o E and EB are re all' o-réc-chio, to say or whisper in any one's ear; a têm-po, has been shown that B A and AC are greater than BE and E C. greater (Ax. 4) than oD and DB. But it in time, in the nick of time; ve-ni-re a grán-di schiê-re, to come Much more then are BA and A c, greater than BD and DC. in great crowds or masses: by the preposition according to (or after); e. g. a ma-niê-ra, after the manner or fashion; a bc-chio, according to a measure taken merely by the eye; a vo-lon-tà di cia-sche-du-no, according to the will or liking of everybody: by the prepositions against or towards; e. g. ri-bel-lar-si ad al-cuno, to rebel or mutiny against somebody; all' orien-te, all occi-den-te, towards the east, west: by the preposition with; e g. a tre cól-pi l' ue cí-se, he killed him with three blows; an-da-re a grán-di pás-si, to walk with long or great steps; stä-re a bócca a-pèr-ta, a oc-chi a-pêr-ti, a brác-cia a- pêr-te, à cá-po chi-no, a chio-me sciôl-te, to stand with an open or gaping mouth, with open arms, with the head inclined, with dishevelled hair; a briglia sciôl-ta, with slackened reins, at full speed or gallop; corri-spón-de-re ad al-cu-no, to agree with somebody; u-ni-to ad alcu-no, united with somebody; pa-ra-go-ná-re ú-na cô-sa a qualche oh-sa, to compare one thing with another: by the preposition for; e. g. con-dan-ná-to a vi-ta dl-le ga-lê-re, condemned B Again, the exterior angle B D C of the triangle C D E is greater than its interior and opposite angle CBD (I. 16). And the exterior angle OEB of the triangle ABE is greater than its interior and opposite angle BAC (I. 16). But the angle BDC is greater than the angle CE B. Much more, therefore, is the angle BDC greater than the angle BAC. Wherefore, if from the ends of, &c. Q. E. D. makes the following observation: "Mons. Clairault, in the Scholium. 1.-Respecting this proposition, Dr. Simson preface to his Elements of Geometry, published in French at Paris, anno 1741, says that Euclid has been at the pains to prove that the two sides of a triangle which is included within another, are together less than the two sides of the triangle which includes it:' but he has forgot to add this condition, viz. that the triangles must be upon the same base: because unless this be added, the sides of the included triangle mav be greater than the sides of the triangle which includes it, in any Take a straight line D E termi- Scholium 2.-Respecting this proposition also, Dr. Thomson very properly remarks, that it "is never referred to by Euclid [in his after writings], except in the eighth proposition of the third Book; and that proposition may be proved without it." The Prop. VIII. Book III. is, in fact, proved without it, in Cassell's Euclid. Thus, it would appear that one of the links of the great Euclidean chain of reasoning is unnecessary. attempts to shorten that chain are no doubt praiseworthy; for art is long, and life is short. Nevertheless, the proposition contains a geometrical truth, useful in many cases of which Euclid never dreamed; and, therefore, we ought to retain this extra link. EXERCISE TO PROPOSITION XXI. All If from any point within a triangle, straight lines be drawn to the vertices of the three angles, these three straight lines taken together shall be less than the sum of the three sides, but greater than half that sum. Let A B E, fig. Y, be a triangle, and c any point in it; also, let straight lines CA, CB, and C E be drawn from c to its three angular points A, B, and E. Then, the sum of the three straight Fig. Y. E B lines CA, CB and CE is less than the sum of the three sides A B, BE and AE; but greater than half their sum. By the preceding proposition, a c and cв together are less than AB and B B together; BC and E c together are less than A B and A B together: and AC and CE together are less than AB and BB together. Therefore, by axiom IV. twice A c, twice BC, and twice c E together, are less than twice A E, twice в E and twice AB together; wherefore, ▲ C, B C and C E together are less than AE, BE and A B together. Again, by Prop. XX., A c and oв together are greater than AB; EC and CE together are greater than BE; and AC and CE together are greater than A E. Therefore, by Axiom IV., twice a C, twice BC, and twice CE together, are greater than A B, BE and AE together. Wherefore, A C, BC, and CE together, are greater than the half of AB, of B E and of A E together. Therefore if from any point within a triangle, &c. Q. E. D. Scholium. In this demonstration, a very obvious axiom is taken for granted, viz. that the halves of unequals are unequal; or, as it may be more explicitly expressed, if the double of one series of magnitudes taken together, be equal to the double of another series of magnitudes taken together, the sum of the former series is equal to the sum of the latter series, &c. PROPOSITION XXII.-PROBLEM. To make a triangle of which the sides shall be equal to three given straight lines, but any two whatever of them must be greater than the third (I. 20). In fig. 22, let A, B, and c be the given straight lines, of which any two whatever are greater than the third, viz., a and B greater than c; A and c greater than B; and B and c greater than A. It is required to make a triangle of which the sides shall be equal to A, B, and c, each to each. HLK. A, B, and c. Because the point is the centre of the circle DXL, the straight line FD is equal (Def. 15) to the straight line FK. But FD is equal (Const.) to the straight line A. Therefore PK is equal (Ax. 1) to A. Again, because a is the centre of the circle LKH, the straight line G H is equal (Def. 15) to the straight line GK. But G H is equal to c. Therefore also a K is equal to c. And re is equal (Const.) to B. Therefore the three straight lines KF, FG, and GK are equal to the three straight lines A, B, and c. Wherefore the triangle KFG has been made, having its three sides KF, FG, and G E, equal to the three given straight lines A, B, and c. Q. E. F. Scholium 1.-This is the general proposition of which Prop. It is evident that upon the other I. is but a particular case. side of the base FG, another triangle might be constructed, In the demonstration, it is assumed that the two circles will having its three sides equal to the three given straight lines. intersect each other. To prove this, it is sufficient to observe that the sum of the radii of the two circles is, by hypothesis, greater than the distance between their centres. Scholium 2.-Dr. Simson remarks on this proposition, that 66 'some authors blame Euclid, because he does not demonstrate that the two circles made use of in the construction of this problem, must cut one another: but this is very plain from the determination he has given, viz. that any two of the straight lines DF, FG, GH, must be greater than the third. For who is so dull, though only beginning to learn the elements, as not to perceive that the circle described from the centre F, fig. 22, at the distance FD, must meet FH betwixt F and H, because FD is less than FH; and that for the like reason, the circle debetwixt D and G; and that these circles must meet one another, cribed from the centre o, at the distance & н, must meet DG because FD and G H are together greater than FG?" Forseeing, however, notwithstanding his predecessor's remarks just cited, that some learners might be so dull as not to perceive what seems so clear to a geometer, Dr. Thomson has very properly inserted the following paragraph in his construction, after the words "another circle HLK;" and to this insertion all beginners would do well to take heed: ference of the circle DKL will cut F H between F and H; and "Now, because (hyp.) F H is greater than FD, the circumtherefore, the circle K H L cannot be wholly within the circle DK L. In like manner, because (hyp.) D G is greater than GH, the circle DKL cannot lie wholly within the circle K H J. Neither can the circles lie wholly without each other, since (hyp.) DF and GH are together greater than FG. The circles must, therefore, intersect each other; let them intersect in the point L." EXERCISE I. TO PROP. XXII. To make a triangle equal to a given triangle. straight lines of which any two are greater than the third, Consider the three sides of the given triangle, as three given Prop. XX., and by the preceding proposition, construct a triangle of which the three sides shall be equal to these three given straight lines. By Prop. VIII. it is plain that the angle angle contained by the two sides equal to them of the other; contained by any two sides of the one triangle is equal to the and by Prop. IV. that the two triangles are equal to each other in all respects. Wherefore, a triangle has been constructed equal to the given triangle. Q. E. F.* *This exercise was solyed by C. L. HADFIELD, Bolton-le-Mooss; Q. PRINGLE, Glasgow; T. BocoOCK, Great Warley; J H. EASTWOOD, Middleton; aud F. J. BREMNER, Carlisle. |