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Ther of "recom" enlarged to double size. The illustration is "irreconciled."

The ring "mis" united to the line "com." The illustration is "misconduct." [The first character should cross the second, and the third should be thin.]

A very short k with a ring, written in a - vowel position. The illustration is "countermand."

A curve with its exterior or conver side crossing the beginning of the next character. The illustration is "extrinsic."

Words.

Marks.

Analysis.

attitude

ut-ut-(yud)

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ca!

col

incal &c.

incalculable

A short instead of a full 7, and the 1 written upwards, to distinguish the combination from cl, as in clever, which would have a downward 1. This contraction is especially useful in words such as "calculate," " colloquy," &c., (the illustrations) where the prefix is followed by a k or a q. There can be no confusion between cal and col, as they are rarely, if ever, followed by the same radical word. The negative prefix, " in " may be indicated by "crossing out" the k-as in the last illus. tration, "incalculable."

or

"un

LESSONS IN LATIN.-No. XXXIX.
By JOHN R. BEARD, D. D.
IRREGULAR VERBS.

WE now come to those verbs which custom characterises as the the Irregular Verbs, inasmuch as they greatly depart from the models supplied in the four conjugations; and first we present Possum.

1. Possum, posse, potui, to be able.

Possum consists of potis, able, and sum, I am. The potis is contracted into the stem pot, and pot before the s in sum, becomes pos; whence comes pos-sum.

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INDICATIVE SUBJUNCTIVE.

PERFECT.

cunque rebus posset patriam javare; Caesar quara potuit maximis itineribus, exercitum contra hostes duxit; casus est, quum sic aliquid evenit, ut vel non evenire vel aliter evenire potuerit; omnes mundi partes ita constitutae sunt, ut neque ad usum meliores potuerunt esse, neque ad speciem pulchriores; ante occupatur vix Caesar milites e castris educere potuerat, quam hostes impetum animus ab iracundiâ, quam providere satis potuit ne occuparetur; fecerunt; quid enumerem artium multitudinem sine quibus vita omnis nulla esse potuisset? quem ut mentiatur inducere possumus (eum) ut pejeret exorare facile poterimus; dolorem si non potero frangère, occultabo; facile intelligitur nec figuram situmque membrorum nostrorum, nec ingenii mentisque vim effici potuisse fortunâ; hoc primum sentio nisi in bonis amicitiam esse non posse. ENGLISH-LATIN.

We cannot conceal wickedness from God; you cannot doubt that the world is governed by a mind; can the world be from nothing? out of nothing, nothing can arise; what can arise out of confused masses? can order arise out of chance? they could not allow good men to be punished; I will return home with the utmost speed; they will return home with the utmost speed; before I could speak I was seized; the world cannot be more beautiful; can those women be more fair? I will give thee a book if I am (shall be) able; he was unable to subdue his grief, but he will be able to conceal it; only among good men can friendship exist; if I could have come I would have told you all; unless they had been able to come we able to take a walk to-morrow; without the arts the life of man should have known nothing; canst thou take a walk? I shall be could not be.

2. Edo, edere, edi, esum, to eat.

This verb in its irregularities has an apparent identity with parts of the verb esse, to be. This arises from the changes I have been able. I may have been, &c. required with regard to sound. The e in sum is short; in the

pot-ui

pot-uistis

pot-uit

pot-uímus

pot-uistis

pot-uerim

pot ueris

pot-uerit

pot-uerĭmus

pot-ueritis

pot-uērunt (ēre) pot-uerint

PLUPERFECT.

I had been able. I might have been,

pot-uĕram

pot-uĕras, &c.

pot-uissem [&c. pot-uisses, &c.

2nd. FUTURE.

I shall have been able.

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No Imperative.

It is thus seen that with the aid of potis the verb is formed by sum and its parts, of which the ui is for fui, the ƒ or aspirate being dropped in combination to prevent the harshness of two consonants coming together, as pot-fui, &c.

VOCABULARY.

Celare (aliquem aliquid) 1, to conceal; enumerare 1, to number; meditari 1, dep. (with acc.) to meditate on; pejĕrare (in its original form perjuro, from per, through; and jus, right), to swear falsely; constituere 3, to appoint, ordain; desistère 3, to stand back from, desist, cease; inducere 3, to lead in, induce; mitescere (no perf., no supine) 3, to become mild, tame; quam potuit maximis itineribus, with the utmost speed; effector, óris, m. a creator; situs, ûs, m. place, position; adeo, to such a degree, greatly; proinde quasi, just as if; casus (cado, I fall), ûs, m. chance.

EXERCISES.-LATIN-ENGLISH.

Pergite, pueri, atque in id studium, in quo estis, incumbite, ut et vobis honori et amicis utilitati et republicae emolumento esse possitis; nemo adeo ferus est ut non mitescère possit; hoc quotidie meditare, ut possis aequo animo vitam relinquere; quidam idcirco Deum esse non putant, quia non apparet nec cernitur; proinde quasi nostram ipsam mentem vidére possímus; universum mundum quum cernimus, possumusne dubitare quin ei praesit aliquis effector et moderator? nihil tam difficile est quin (=ut non) quaerendo investigari posset; sic cogitandum est tanquam aliquis in pectus intimum inspicere possit, et potest; satis nobis persuasum esse debet, etiamsi Deum hominesque celare possimus, nihil tamen injuste esse faciendum; potestisne dubitare quin Deus universum mundum gubernet? non possumus; cur nobiscum ambulare non potes? Alcibiades Athenas Lacedaemóniis servire non poterat pati. Marcellus pedites primum, deinde equites, quanto maximo possent' impetu in hostem erumpere jussit; Agesilaus non destitit quibus'

parts of edere it is long, inasmuch as it involves a contraction. PRESENT INDIC.: Edo, edis (es), edit (est), edimus, editis (estis), edunt. IMPERF. SUBJUNC: Ederem (essem), ederes (esses), ederet (esset), ederemus (essemus), ederetis (essetis), ederent (essent). IMPERATIVE: Ede (es), edite (este),

edito (esto), editote (estote), edunto.

The other parts are regular; only for editur, estur is found; and for ambedens, ambens, eating round. So the compounds comedo, comědis, comēs (to eat up), exedo, exedis, exēs (to eat up or out). Comedo has comesus, as well as comestus.

VOCABULARY.

Symbola, ae, a contribution; de symbolis esse, to eat at the common expense, to enjoy a pic-nic; adolescentulus, i, a young man; argentum vivum, quicksilver; curculio, ónis, the corn worm; moles, is, f. a mass; familiaris, e, belonging to the family; res familiaris, property, substance; perrumpere 3, to break through, break into; vae, woe! alas! modicé, moderately.

EXERCISES.-LATIN-ENGLISH.

Esse oportet ut vivamus; non, vivere ut edamus; modice bibite et este; heri aliquot adolescentuli convenerunt ut de symbolis essent; haec herba acerba esu est; aegritudo lacerat, exest animum planeque conficit; curculiones frumentum exesse incipiunt; argentum vivum exest ac perrumpit vasa; majores nostri cavere non potuerunt, ne vetustas monumenta exesset; quae unquam moles tam firma fuit quam non exessent undae? vae vobis qui omnem rem familiarem luxuriâ comestis! fabulae narrant Saturnum liberos ex se natos comesse solitum esse; consumit enim aetas temporum spatia.

ENGLISH-LATIN.

Saturn did not devour his children do you think that Saturn devoured his children? the waves eat away rocks; thou livest to eat; thou oughtest to eat in order to live; they eat very little; we are going into the country in order to enjoy a pic-nic; this bread old is bitter to eat; corn-worms have eaten up the corn; age devours all things; grief will devour the mind, and destroy life. they have eaten and drunk moderately; a wise man will eat little 3. Fero, ferre, tuli, latum, to bear. PRESENT ACTIVE.

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The other parts are regularly formed from fero, tuli and latum; as Subj. Pres., feram, as, at, &c.; ferar, aris, atur, &c.; Ind. Imp., ferebam, and ferebar; Fut., feram, es, et, &c., ferar, éris, étur, &c.; Subj. Perf., tulerim, is, it, &c.; Plup. Ind., tuleram, &c.; Plup. Subj., tulissem, &c.; Inf. Fut., laturum esse; Part. Pres., ferens; Part. Fut. laturus, a, um; Part. Pass., latus; Part. Pass., in dus, ferendus; Gerund, ferundum.

So also the compounds, as offero, obtuli, oblatum, to bring before. From the stem of the perfect tuli, arose tollo, tollere, sus-tali, sub-latum, to raise, take away.

VOCABULARY.

Affero, afferre, attuli, allatum, to bring to; aufero (ab and fero), auferre, abs-tuli, ablatum, to take away; confero, conferre, contuli, collatum, to bring together, contribute, compare; defero, deferre, detuli, delatum, to bring down, present, accuse; effero, efferre, extuli, elatum, to bring out, to carry out for interment, to bury; infero, inferre, intuli, illatum, to bring in; bellum infero alicui, to make war on; praefero, praeferre, praetuli, praelatum, to bring before, prefer; refero, referre, retuli, relatum, to bring back, report, refer; hoc est, with genitive, this is the sign, or, character of; decedere, to depart, die; doctor, oris, m. a teacher; gigas, gigantis, m. a giant; commodum, i, n. advantage, convenience; funditus, from the foundation, thoroughly; qui (quo), whereof, whereby.

The compounds of fero are a good study in relation to the forms which prepositions take in combination, and the modifications of meaning which they occasion.

EXERCISES.-LATIN-ENGLISH.

Ferte misero atque inopi auxilium; confer nostram longissimam actatem cum aeternitate, et brevissima videbitur; quid quaeque nox aut dies ferat, incertum est; incumbe in eam curam et cogitationem, quae tibi summam dignitatem et gloriam afferat; ferre laborem consuetudo docet; pecuniam praeferre amicitiae sordidum est; ut quisque maxime ad suum commodum refert, quaecunque agit ita minime est vir bonus; bonum civem reipublicae dignitatem suis omnibus commodis praeferre oportet; hoc doctoris intelligentis est videre quo ferat natura sua quemque; Aristides in tantâ paupertate decessit ut qui efferretur, vix reliquerit; poetae ferunt gigantes bellum diis intulisse; Socrates eundem vultum domum referebat quem domo extulerat; quod auri, quod argenti, quod ornamentorum in urbibus Siciliae fuit id Verres abstulit; multi etiam naturae vitium meditatione atque exercitatione sustulerunt; pietate adversus Deum sublata, fides etiam et societas humani generis tollitur; qui Deum esse negant, nonne omnem religionem funditus sustulerunt? Caritate benevolentiâ que sublatâ, omnis est e vitâ sublata jucunditas.

ENGLISH-LATIN.

Compare thy folly with thy father's wisdom; I have compared my sin with God's love; I will compare small things with great; I have borne a mass of evil; a mass of evil has been borne by me; the giants are said to have raised mountains; I know not what the day may bear (bring); bear the labour patiently (with an equal mind); do not refer all things to thy own advantage; the enemy has taken away what gold and silver I had; love being taken away, all the pleasure of home is taken away; canst thou take away the fault of nature by meditation? do not take away the faith and intercourse of life.

Volo, volle, volui, to be willing, to wish. 4 Nōlo, nolle, nolui, to be unwilling, refuse. Malo, malle, malui, to be more willing, prefer. Nolo is made up of non and volo; as, non-volo, nolo; and malo is made up of magis and volo; as, magis-volo, mavolo. Consequently, the first vowel of nōlo and malo is long, while that of volo is short.

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IMPERATIVE of volo and malo none).

8. 2. Nōli, nolito; 3, nolito; P. 2, nolite, nolitāte; 3, nolunto. PARTICIPLE.

nolens, tis;

Volens, tis; (of malo none.) The forms that are made from the perfect are regular, thus: volui, voluerim, voluero, volueram, voluisse, voluissem. The other parts are wanting. VOCABULARY.

Defatigare, to weary, be weary; nobilitare, to make known, or celebrated; publicare, to make public; sectari (with acc.), to follow, strive after; adstringere, to bind; serius, a, um, earnest, serious; ejusmodi, of that kind; acer, acris, acre, sharp, energetic; velim nolim, whether I will or not, will-I, nill-I; faber, ri, a workman; faber lignarius, a carpenter.

EXERCISES.-LATIN-ENGLISH.

Qui virtutem suam publicari vult, non virtuti labórat sed gloriae; nonne poetae post mortem nobilitari volunt? ego non eadem volo senex quae volui adolescens; si vis amari, ama; bono mentis fruendum est, si beati esse volumus; docilis est qui attente vult audire; omnia benefacta in luce se collocari volunt; si acres ac diligentes esse vultis, magna saepe intelligetis ex parvis; quem docilem velis facere, simul attentum facias oportet; sic cum inferipraeclare Socrates hanc viam ad gloriam proximam dicebat esse, si ore vivamus, quemadmodum nobiscum superiorem velimus vivere; quis id ageret ut qualis haberi vellet, talis esset; si quis veram gloriam adipisci volet, virtutis officiis fungi debebit; nolumus in bus rebus excellere; si quid per jocum dixi, nolito in serium conservandis bonis viris defatigari; homines nolunt eundem pluriconvertere; libero sum judicio, nulla ejusmodi adstrictus necessitate ut mihi, velim nolim, sit certa tuenda sententia; Socrates noluit ex carcere educi quum facile posset; ego me Phidiam esse mallem quam vel optimum fabrum lignarium; Utrum corporis an ingenii vires tibi augeri mavis? virtute in aliâ alius vult excellere; quibus id persuasum est ut nihil malint se esse quam bonos viros, is reliqua facilis est doctrina; vae vobis qui divitias quam virtutem sectari mavultis; malumus cum virtute paucis contenti esse, quam sine virtute multa habere; Aristides, Atheniensis, bonus esse malebat quam videri. ENGLISH-LATIN.

riches; do not wish to excel in luxury; I wish to excel in virtue; They wish to be wise; they prefer to have wisdom rather than dost thou wish to take a walk with me? I would rather read this book; they refused to go from their homes; he will refuse to hear what thou wishest to say; if any one shall wish to become wise, let him read the best books; men are unwilling for the same person to than riches; I prefer to be wise than to be rich; few prefer have learning, riches, and power; I would rather have learning wisdom to power.

Aesopii Fabulae.

VIATORES ET ASINUS.

spicati, accurrunt laeti, et uterque eum sibi vindicare coepit, quod eum prior conspexisset. Dum vero contendunt et rixantur, nec a verberibus abstinent, asinus aufugit, et neuter eo potitur.

Duo qui una iter faciebant, asinum oberrantem in solitudine con

CORVUS ET LUPI.

Corvus partem praedae petebat a lupis, quod eos totum diens comitatus esset. Cui illi, "Non tu nos," inquiunt," sed praedamı secatus es, idque eo animo, ut ne nostris quidem corporibus parceres, si exanimarentur." in actionibus non spectatur quid fiat sed quo animo fiat. PASTORES ET LUPUS.

tores caesâ ove convivium celebrabant. Quod quum lupus cerneret, "ego," inquit, "si agnum rapuissem, quantus tumultus fieret! At isti impune ovem comědunt!" Tum unus pastorum, "nos enim," inquit, "nostrâ non alienâ ove epulamur." TUBICEN.

Tubicen ab hostibus captus, "ne me," inquit, "interficite, nan inermis sum, neque quidquam habeo praeter hanc tubam." At hostes, "Propter hoc ipsum," inquiunt, "te interimemus, quod quum ipse pugnandi sis imperītus, alios ad pugnam incitare soles."-Fabula docet non solum maleficos esse puniendos, sed etiam eos qui alios ad male faciendum irrītent.

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columbas converterunt.-Haec fabula docet potentiorum discordias be considered as prolonged to any extent a motive may ever arise imbecillioribus saepe prodesse.

VOCABULARY.

Iter facere, to travel; una, together; oberro 1, I stray; sibi vindico, I claim; aufugio (ab and fugio) 3, I flee; viator, óris, m. a traveller; corvus, i, a raven; exanimor 1, to be exhausted, die; epulor 1 (dep.), Ifeast on (with abl.); tubicen, inis, m. a trumpeter; interimo 3, I kill; irrito 1, to provoke, move, induce (quâ refers to pax).

LESSONS IN GEOMETRY.-No. XIV.
LECTURES ON EUCLID.

In our last lesson, we brought before the notice of our students
and readers, the fact of the existence of a "Geometry without
Axioms," and made some useful extracts from that curious and
ingenious work, particularly from the "Intercalary Book," in
which the author demonstrates the truth of the axioms and
postulates of Euclid. We should have been glad to continue
these extracts for the benefit of our students, but they are too
extensive, and perhaps rather too learned for the generality of
our readers; we shall, therefore, content ourselves with giving
the enunciations of the propositions and corollaries of the rest
of the "Intercalary Book," recommending the whole work
strongly to their attention as containing more ingenious and
original thought in reference to the matters contained in the
"First Book" of Euclid, than we have ever seen in any treatise
or commentary on that book.

Prop. 2. Cor.: Any solid, surface, line, or figure, may be turned about any one point, or about any two points; such point or points

remaining unmoved.

for desiring. If a given straight line in a plane be turned in that plane about one of its extremities which remains at rest, till the straight line is returned to the situation from which it set out; the plane figure described by such straight line is called a circle, and its boundary the circumference. The points in which one extremity of the straight line remains at rest, is cailed the centre of the circle. Any straight line from the centre of a circle to the circumference, is called a radius of the circle; and any straight line passing through the centre and terminated both ways by the circumference, is called a diameter of the circle. When a circle is said to be described about a given centre with a given straight line for radius, the meaning is, that it is described by the revolution of the given straight line about one of its extremities.

Prop. 12. Cor. 1: A plane surface may be prolonged to any extent in a plane.

Prop. 12. Cor. 2: If a straight line in a plane be prolonged, its prolongation lies wholly in that plane.

Prop. 12. Cor. 3: A circle may be described about any centre, and with any radius.

Prop. 12. Cor. 4: All the radii of the same circle are equal. And circles that have equal radii, are equal.

Prop. 12. Cor. 5: Any straight line from the centre of a circle to a point inside being prolonged, cuts the circumference in a point only. Prop. 13: Any three points are in the same plane. [That is to say, one plane may be made to pass through them all.] Prop. 13. Cor. 1: Any three points which are not in the same straight line being joined, the straight lines which are the sides of the three-sided figure that is formed, lie all in one plane.

Prop. 13. Cor. 2: Any two straight lines which proceed from the same point, lie wholly in one plane.

the same straight line) are made to coincide with three points in Prop. 13. Cor. 3: If three points in one plane (which are not in another plane, the planes shall coincide throughout, to any extent to which they may be prolonged.

Prop 8: From one of two assigned points to the other, may be described a line, which being turned about its extreme points, every point in it shall be without change of place. Such a line is called a straight line. A body or figure which is turned about two points in it that are also the extremities of a straight line (inas-repeat Euclid's Demonstrations in full, seeing that we have much as the whole of the straight line remains without change of place), is said to be turned round such straight line. When from any point to any other point, a straight line is described or made to pass; the two points are said to be joined. If to a straight line addition is made at either end, in such manner that the whole continues to be a straight line, the original straight line is said to be prolonged, and the part added is called its prolongation."

Prop. 8. Cor. A straight line may be described or made to pass from any one point to any other point.

Prop. 9: Between two points there cannot be more than one straight line.

Prop. 9. Cor. 1: Two straight lines cannot inclose a space. Prop. 9. Cor. 2: Any portion of a straight line is also a straight line.

Prop. 10: Two straight lines, which are not in one and the same line, cannot have a common segment.

Prop. 10. Cor. 1: If any two points in one straight line coincide with two points in another, the two straight lines shall coincide with one another to the extent of the length that is common to both, and be one and the same straight line throughout.

Prop. 10. Cor. 3: Any three points (which are not in the same straight line)' being joined, there shall be formed a three-sided figure; and no point in any one side shall coincide with any one point in either of the other sides, except the points which were joined.

Prop. 10. Cor. 4: If two straight lines cut one another, they coincide only in a point.

Prop. 10. Cor. 5: Any straight line may be applied to any other, so that they shall coincide to the extent of the length common to

both.

Prop. 10. Cor. 8: A terminated straight line may be prolonged to any length in a straight line. If to a plane surface, addition is made in any direction, in such manner that the whole continues to be a plane surface, the original plane surface is said to be prolonged, and the part added is called its prolongation.

Prop. 12: A surface may be described in which any two points being taken, the straight line between them lies wholly in that | surface. Such a surface is called a plane surface; or when no particular boundaries to it are intended to be specified, a plane. A figure which lies wholly in one plane, is called a plane figure. The whole plane surface within the boundaries of a plane figure which is bounded on all sides, is called the area of the figure; and its whole linear boundary, of whatever kind or composition, is called the perimeter. When a straight line is said to be of unlimited length, the meaning is, that no point is assigned at which it shall be held to be terminated, but on the contrary it shall without further notice

We now proceed to explain the Propositions of the First Book of Euclid in proper order. In so doing, we shall not already published these in Cassell's edition (the cheapest by far that ever was given to the public); but we shall give either an abridgement of them in a form that will be useful to learners and beginners, or else a new demonstration of them altogether; especially, in those cases where the old is difficult, circuitous,

or tedious. We shall then solve or demonstrate all the exer

cises appended to each proposition in Cassell's edition; where the corollaries in that work require either demonstration or elucidation, it shall be given; and where practical illustrations can be introduced with facility, it shall be done.

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From the point A as a centre, and with the straight line AB as a radius, describe the circle B CD. From the point в as a centre, and with the same straight line BA as a radius, describe the circle ACE. Then, from the point c, draw straight lines to the points A and B, that is, join CA and CB; and the figure ABC having the three sides A C, C B, and B A, is an equilateral triangle; in other words, these three sides are equal to each other.

The reason of this is very plain; by the definition of a circle, all its radii are equal (Def. 15). Therefore, the different radii AC and B C, are each equal to the common radius A B. Wherefore (by Axiom I.), the two radii A c and в C, are equal to each other. It follows, therefore, that all the three radii A B, A C, BC, are equal to each other, and the triangle is equilateral (Def. 24).

In the demonstration of this proposition it is taken for granted that the two circles BCD and AC E must intersect each other. This case is included in the general proposition, that if the radii of two circles, and the distance between them be such, that of the straight lines which represent them, or which are equal to them respectively, any two are greater than the th these circles must intersect each other. The demonstrat this proposition is generally either appended to the 22nd

case,

banks are represented by the lines mn and op. If we place an equilateral triangle r (say, formed of wood) horizontally, at the point B, so that by looking along the first edge of it, we see the point A, and along the second edge of it, the point D; then, by transferring it to another point, c, in the direction of the straight line BD, and placing it horizontally at that point, we see the point a, along a third edge of it, and the point B, backwards, along the second edge of it; we are then assured that the figure A B C is an equilateral triangle, and that the distance BC is the same as the distance BA or AC; we have, therefore, only to measure the accessible distance BC, and instantly we know not only the inaccessible distance BA, which was required, but also the inaccessible distance cA. This method may be so varied as to measure perpendicular distances as well as those that are inclined at any angle whatever. The adjustments necessary for this purpose, will readily suggest themselves to the ingenious student.

of Book I., or is included in its demonstration. In the present | accessible by reason of a river running between them, whose if the straight line AB be produced both ways to meet the circumferences of the two circles in the points D and E, it is plain that because A E is greater than AD, the circumference of the circle BCD will cut A E between a and E, and therefore the circle A CE cannot lie wholly within the circle B CD. Also, because D B is greater than B E, the circle B C D cannot lie wholly within the circle A CE. Neither can the two circles be wholly without each other, because AD and BE are together greater than A B. Therefore, the two circles must intersect each other. From the construction of this problem, it is plain that upon any given straight line, there can be described two separate equilateral triangles, namely, one on each side of the straight line; because the circles intersect each other in two opposite points, c and F; where c is the vertex of the triangle ABC, and F the vertex of another triangle which would be formed by joining FA and F B. The figure formed of these two triangles would be a rhombus, but this cannot yet be mathematically demonstrated, although sufficiently obvious to the drawer of the figures. Neither can it yet be proved that the two circles must cut each other in two points, and only in two. The exercise upon this proposition in Cassell's edition, viz., "To describe an isosceles triangle upon a given finite straight line, that shall have each of its sides double the base," has been already solved in Vol. I., p. 110, col. 1. Here we may add another which cannot be demonstrated, until we reach the 32nd proposition of Book I. In fig. 1, suppose C F to be joined; then let D C and DF be joined; and the triangle CDF shall be equilateral. Also, let EC and E F be joined; and the triangle ECF shall be equilateral, and equal to the triangle

CDF.

SCHOLIUM 1.-It is obvious enough to any one that considers the nature of an equilateral triangle, that since the three sides are equal to one another, the three angles must be equal to one another, on the principle of the sufficient reason; viz. that whatever reason may be assigned, from the nature of the sides, to show that any two angles are unequal, the same reason may be assigned to show that any other two angles are unequal; therefore, all the three angles must be equal to one another. The fact that they are equal, however, follows as a natural consequence, from the demonstration of the fifth proposition of Book I. We shall, therefore, not forestall our reasoning much, by assuming that if the three sides of a triangle be equal to one another, the three angles of that triangle are equal to one another. The converse of this proposition is equally true, and is a natural consequence of the eighth proposition of Book I., viz. that if the three angles of a triangle are equal to one another, the three sides are also equal to one another. The principle of the sufficient reason will apply in this case. It may also be at once inferred that all equilateral triangles are similar, that is, have their angles equal, and their sides proportional.

SCHOLIUM 2. The preceding principles being admitted as legitimate, we have a practical method, derivable from them, of measuring the length of any inaccessible distance. Suppose, for instance, that in fig. (A) we wished to know the

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LESSONS IN FRENCH.-No. XLII.
By Professor LOUIS FAEQUELLE, LL.D.
SECTION XCVII.

PRACTICAL RÉSUMÉ OF THE Rules on the past pARTICIPLE.—I.
The participle past is VARIABLE under any of the following
conditions:-

1. When employed as an adjective; in which case it agrees in gender and number with the noun which it qualifies:Des livres imprimés. Ces femmes paraissent bien abat- Those women appear very dejected,

tues.

Printed books.

[blocks in formation]

6. When used along with être (as in Rule 5) in the formation of the compound tenses of those reflective verbs, in which the reflective pronoun is not the direct, but the indirect object of the proposition; in which event it agrees with the direct object, provided (as in Rule 4) that object precedes it :

Les histoires qu'elles se sont racon- The stories which they related to each
tées.
other.

7. When forming part of a compound tense of a verb governing a succeeding infinitive, it is at the same time preceded by a direct object which is represented as performing the action denoted by the infinitive; in which condition it agrees with that direct object :

Les dames que j'ai entendues chan- The ladies whom I heard sing (singing).

ter.

8. When in a sentence containing the pronoun en, the participle is preceded by another object or regimen which is direct; in which case it agrees with that direct object:

Je les en ai avertis.

distance between two points, A and B, which are rendered in- Vous les en avez informés.

I have warned them of it.
You have informed them of it.

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