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crater, these fallen materials formed strata inclining outwards, as represented in fig. 32. The section in fig. 32 represents an imaginary restoration of the island as represented in fig. 31. The dotted lines denote the upper parts of the cone, which surrounded the crater in the centre. All those parts, which were once visible, have been washed away by the denuding action of the sea. The strong lines represent the parts of the island, which are still under water. In the centre is a dyke of solid lava, one

hundred feet in diameter, which has cooled in the vent of the crater, where it was once in a fused state. It is this solid lava that now forms the reef already mentioned as being eleven feet under water. This has withstood the action of the waves, while all the loose materials around it, represented by the dotted lines, have been washed away to a lower level. The beds which have settled on it are formed some of sand, and some of stones and volcanic fragments. The stones which were ejected never exceeded a foot in diameter, and consisted of dolomitic limestone. At no time did the island present the appearance of any overflowing of lava from the crater. It is, nevertheless, possible that melted matter may have flowed out

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from the flames of the cone, at some depth under water, as is often the case on land, and this flowing of lava may have spread in a broad sheet over the bed of the ocean, and enveloped some Mediterranean fish. This reef furnishes an instance of what has been regarded by some as a geological aradox, in which the lava, which was the lowest part of the island while rising, and a part which scarcely rose to the level of the sea, is now become the highest point of the land. These two islands, Sabrina, in the Azores, and Graham, in the Mediterranean, as being the most modern and the best examined, are instances sufficient to explain to you the phenomena of submarine volcanoes. I would, otherwise, have referred you to the rise of an island called Nyée, on the coast of Iceland, during the tremendous eruption of Skaptar Joku), which occurred in 1783; an island which afterwards disappeared: and to another which, in the form of a peak, with some low

conical hills upon it, rose from the sea a little to the east of Kamtschatka. This island still remains. Islands of this description are not always raised to their full elevation by a single paroxysm of volcanic power, but are raised by a succession of efforts which are repeated for months and years. Though the surfaces of these islands are formed much by the volcanic materials which are thrown up and fall again, there are instances in which they consist mainly of the rocks which had formed the bottom of the sea, the beds of which have been upheaved by the power from below. This is the case with an island called New Kameni, near Santorin, in the Grecian Archipelago, which was raised up in 1707, and is food in great measure of limestone covered with. shells.

L ES SONS IN G E R MAN.—No. XV SECTION XXVIII.

380 refers to the place where anything may be supposed to erist or transpire. Ex.: "Be ist mein oleffor? Where is my knife? *Bo taufen tie Rinter? Where (in what place) are the children running 2 Øa is used in answer to me; that is, to designate some particular place; as, oa is es; ta (aufen fit. §in denotes direction, or motion from the speaker; as, QBatum taufen tie Rinter bino Why are the children running thither 2 §er is the opposite, in signification, to him; denoting motion or direction toward the speaker; as, IBarum taufen tie Rinter str? Why are the children running hither? §ier signifies “in this place;” as, QBatum bleiben tie Rinter hier? Why do the children remain here 2 These words are frequently compounded, one with the other; thus, from mo and him; we have the compound luchin; from no and her, mucher; from ta and him, talin; from tu and her, tuhtt, from bitt and him, hierhill; and from Wirt and her, hierotr (sometimes contracted to hi.jer). § 103. 3.

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1. Where is the picture-gallery of this town 2. Where was that gentleman born”? 3. He was born in Bohemia. 4. Where does your friend, the actor, reside 5. He resides in the city. 6. Whither do these emigrants go? 7. Whence do these immigrants come * 8. They come from France. 9. Where much is given, much is required. 10. Here the revenge” and whetted sword" of a traitor enter not”;-beneath” the shade of this tree comes no king 11. He threw down the book before me. 12. Whither art thou going? 13. I am going to my brother-in-law. 14. Will these emigrants go to America. 15. No, they will stop here. 16. There is water in the pond. 17. Where does she come from ? 18. She comes from Germany.

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1. When did he live? 2. He lived in the fourteenth century". 3. My friend told me he would never go there again”. 4. Do you go to Spain? 5. No, I shall not go thither. 6. The commander-in-chief has sent his troops where the danger was most". 7. Is this ship from Spain or from Havre 8. No, it is neither" from Spain nor” from Havre; it comes from Hamburg. 9. These immigrants are going to Milwaukee, and are emigrants from Bohemia and Wenedig. 10, Can you leap over that gate"? 11. I could when I was young. 12. He hade" me to go whither that he might speak to me about it.

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XIII. [A term or boundary is the extremity of any thing.] This definition is unnecessary, being merely verbal. xiv. A fgure is that which is enclosed by one or more boundaries. nition is applicable to solid figures, as well as to plane figures. xV. A circle is a plane figure contained [or, bounded] by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. The circumference of a circle is its boundary. The space contained within the boundary, is called the circle.

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XVI.

And this point is called the centre of the circle. The straight line drawn from the centre to the circumference, is called the radius. * xvii. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. XVIII A semicircle is the figure contained by a diameter and the part of the cir cumference cut off by the diameter.

From the 14th definition it will appear that the diagrams of the preceding definitions, as well as those of a few subsequent definitions, and of many propositions in Euclid, are not to be considered as figures in the geometrical sense of the word. The common meaning of the term figure is, according to Johnson, as used by Addison, “something formed in resemblance of somewhat else.” This definition applies to the use of the term figure as denoting diagrams, representations, or drawings of every kind; but it does not apply to the ideas formed in the mind of what are called geometrical figures, which are entirely creatures of the understanding.

With regard to the definition of a circle this has already been explained at some length in Lesson VIII, p. 277, and, therefore, it will only be necessary to advert to some additional particulars of importance relating to the circle, We find that geometers in general are not sufficiently precise in speaking of the circle; very often when they use this term they mean only the circumference of the circle, i.e., the outward boundary exclusive of the space contained within it. Now by the ... as defined in the 15th definition, is meant the space contained within the boundary, and the boundary too; for we cannot conceive of space of a certain form without including its outline or contour in the idea. But we can conceive of the boundary of a circle, without the space contained within it, just as we can conceive of a ring without the finger which fills up the space within the ring. It is better, therefore, when the space is included in the idea of a circle, to use the term circle only; but when the outline or circumference only is meant to say the circumference of the circle. Here it may be added, that any part of the circumference of a circle is called an are, a term evidently borrowed from its likeness to a bow, which in Latin is called arcus. In like manner, carrying out the analogy, the straight line which joins the extremities of an are is called a chord, from the Latin chorda, a string ; while the space incl.-led within the are and its chord is called a segment, i.e., a cro’ing, from the Latin secare, to cut. As to the term radius, it is also taken from the Latin, in which it first signified a rod or staff, and then the spoke of a wheel, whence it came to be applied to the straight line drawn from the centre to the circumference. Then, as to the term diameter, this comes from the Greek, and signifies a measuring off or through, that is, the greatest measure of the circle, being right across the centre, which gives the fullest or truest measure of the circle; it is also applied to the straight line which joins the extreme points of any figure; and hence, metaphorically, to points or questions in metaphysics, which are exactly opposite to each other. Hence, the meaning of the emphatic expression, diametrically opposite to each other. From their definition it is plain that the diameter of a circle is double of its radius, But the question may be asked—What is the ratio or proportion of the circumference to the diameter: . At first sight, this question seems quite absurd ; for what ratio can possibly exist between two things that are perfectly unlike? the dia. meter of a circle is a straight line ; but the circumference of a circle is a curve line; therefore it does not appear that we can compare two such unlike things together, that is, they have no ratio to each other; and, taking them as they actually exist in the circle, this is perfectly true. But if we say, supposing that the circumference of a circle were stretched out in a straight line to its full extent, just as we should do with a wire ring, by cutting it through in one part, and then stretching it out into a straight piece of wire; what would then be the proportion of the circumference thus stretched out, to the diameter of the circle? The question is easily put, but not so easily answered. Mechanics have, in all ages, fancied that they could reach the determination of this proportion by mechanical means; but from the days of Archimedes until now, they have signally failed; and that failure is simply owing to the nature of the thing. We can never compare the physical with the ideal; the mechanical operations of the finest aris, with the pure and simple abstractions of the mind. Even in the fine arts, there is an ideal beauty which no artist or connoisseur could ever reach; so in the mathematical sciences, there is an accuracy of ratio or proportion, which the finest instruments ever constructed, or ever to be constructed by man, can never attain. But ideas can reach it; mathematical expression can reach it; and yet it surpasses the simply arithmetical and practical. We can only approximate to this proportion by ordinary, numbers; for, it is positively an infinite series of which the law cannot be exressed in the ordinary terms of the decimal notation. It is etween 3}} and 3}} as found by Archimedes; that is, the circumference of a circle is less than 3}} times the diameter and greater than 3} times the diameter. As 3}} is the same as 3} or *, it is usually said that theratio of the circumference to the diameter is that of 3} to 1, or of 22 to 7 ; in other words, that if the diameter of a circle be 1 foot, its circumference will be 3} feet; or if the diameter be 7 feet, the circumference will be 22 feet. This approximation is very well adapted for rough calculations, but it is not sufficiently accurate for many important purposes. It gives the circumference too large by rather more than 1 part when the diameter is divided into 800 parts; that is, if the diameter of a circle were 800 feet, the circumference found from the Archimedean proportion, would be rather more than 1 foot too much. Many persons think that it is a very easy matter to determine this ratio. For instance, suppose, say they, that we take the crown of a hat, and measure it right across; well, perhaps this is 7 inches. Now take a string or tape and measure it round; well, perhaps this is found to be 22 inches. Here, then, is the

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proportion in question, what more do you want? it must be right, for we have measured. But, my good friends, measure as often as you like, and take more exact things than the crown of the hat, if you will; still this ratio is erroneous to the extent that we have said. And mind, that a small error in the circumference, leads to a greater error in the surface or area of the circle; and to still greater error in the cubic content of the sphere; for errors are increased by multiplication, Seeing the necessity of being more accurate in regard to the ratio of the diameter to the circumference, Metius, a Dutch mathematician of the 17th century, discovered the proportion 113: 355; that is, the diameter of a circle is to its circumference, very nearly as 113 is to 355. Besides its close approximation to the truth, this ratio is also easily remembered; for we have only to write the first three odd numbers in pairs, thus: 113355 and then divide them into triplets, thus, 113: 355, and we have the ratio in question. The Hindoos, however, appear to have been early acquainted with a convenient approximation much used by us, in the present day. In the Ayeen Akberry, or Institutes of Akbar, the diameter is said to be to the circumference as 1250 is to 3927; now if we multiply both terms of this ratio by 8, which does not alter its value or meaning, we have that of 10000 to 31416, or decimally that of 1 to 3.1416,a very commonly used proportion. The meaning of this proportion is that the circumference is equal to 3 times the diameter and a part of the diameter, additional to this three times, denoted by the fraction of or l'o', ; this fraction again, means that if the diameter were divided into 10000 parts, we must add 1416 of these parts to 3 times its length to get the length of the circumference; or if the diameter were divided into 1250 parts, we must add 177 of these parts to three times its length to get the length of the circumference. The proportion of Metius, however, is really more accurate than the preceding; for it states that the circumference is 3 times the diameter, and a part of the diameter denoted by or ; that is, if the diameter were divided into 113 parts, we must add 16 of these parts to three times its length to get the circumference. The following list of proportions will show the most important and useful approximations to the truth, the last being true to the lowest figure of the decimal, but capable of indefinite extension.

Authors. Original Ratios. The Same in Decimals. Archimedes 7 : 22 I ; 3. 1428.57142857 Lambert 106 : 333 1 : 3' 1415094.33962 Hindoos 1250 : 3927 1 : 3.141600000000 Metius 113: 355 1: 3.1415929203534Lambert 33102 : 103993 1: 3-141592653011+ Lambert 33215: 104348 1: 3-141592653921-1Lambert 66317: 208:341 1: 3.141592653467–H Wallis 1:2 (; ; ; ; ; ; ; &c.) 1: 8-141592653580+

In the latter ratio, that of Wallis, the series continues to infinity, the numerators of the fractions in the parenthesis being the series of the squares of the even numbers, and the denominators being the series of odd numbers reduplicated as factors in the denominators in the manner shown. The fractions in this series are continually approaching to the value 1, or unity. By comparing these ratios as reduced to their decimal values with the last, which is correct as far as it goes, the amount of error in each can be duly appreciated. Thus, in the case of that of Metius, it will be seen that his ratio makes the circumference too great by a little more than 1, or unity, when the diameter is 4 millions. This is what we call “being down upon the eircle;” but we must delay the rest of our lesson on this subject till our next opportunity.

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The four elements of the ancients were fire, air, earth, and water.

-I have chosen to write my poem (annus mirabilis) in quafrains or stanzas of four in alternate rhyme, because I have ever judged them more noble and of greater dignity both for the sound and number than

any other verse in use amongst us."-Dryden. Quinque (quint), Latin, fee, occurs in quinquennial (annus, Lat. a year), happening every five years; in quintessence (essentia, Lat. essence); and in quintuple, fivefold. “Aristoteles of Stagira hath put down for principles these three, to wit, a certain forme called entelechia, matter, [and] privation: for elements, four; and for a fifth, quintessence, the heavenly body which is immutable.”—Holland, “Plutarch.” Re(red), of Latin origin, primarily signifies oek, backward (and has nothing to do with ere, nor does it mean before, as Richardson states, as return, to turn back; hence opposition, as resist, to stand against; also repetition, as revive, to live again; reform, to make again. Re, denoting back:“To desire there were no God were plainly to unwish their own being. which must needs be annihilated in the subtraction of that essence which substantially supported them, and restrains them from regression into nothing.”-Browne, “vulgar Errors.” Re, denoting opposition:“To this sweet voyce a dainty musique fitted its well-tuned strings, and to her notes consorted; And while with skilful voice the song she dittied. The blabbing echo had her words retorted." Spenser. Re, denoting repetition, as in rehearse, recapitulate, remove, No. 1 - The land of silence and of death Attends my next remove." Morts. Resometimes merely strengthens the word, as in receive, reception capio, Lat: I take); and recommend (mando, from manus, a and 1 and do, I wive). - nest, of Latin origin (rectus, straight), appears in rectory (facio, Lat, i make), to make strawht, in rectangular (angulus, Lat a cor.." right-angled; rectilinear (linea, Lal. a line), straighs-lined; and rectitude, uprightness. neiro, Latin, backword, as in retrogradation (gradior, Lat. 1 wait), going backward. It is found, also, in retroactive (ago, Lal, I do, act), acting on a backward direction * A will of palus and penalties was introduced, a retrove statute, to punion the dawaces which did not exist at the time they were coulinitied."—thlbow, "Mewlra." se, of Latin origin, the base of sine, without, denotes tion; apart, from, without , as, seclude (clawdo. Lat: I sowt), to shut out, secede (cello, I go, wield), to withdraw from; seduce (dugo, I lead), to lead from duty. “From the fine gold I separate the allay And show how hasty writers sometimes stray.” Dryden, “Art of Poetry.” Sept, of Latin origin (septem, seven), appears in septennial (annus), occurring every seven years; and in septentrion, the seven stars; the great Bear; Charles's Wain, the north. “Thou art as opposite to every good As the antipodes are unto us, or as the South to the Septentrion.” Shakspeare, “Hen. VI." (3rd pt.) ser, Latin, sir, is found in serangular, six-angled; serennial, every six years; sextuple, sixfold; seragenary, threescore, &c. - these are the seragenary fair ones, who, whether they were handsome or not in the last century, ought at least in this to reduce themselves to a decency of dress suitable to their years."—Chesterfield, "Comnon-sense.” son, of Latin origin (solus, alone), is seen in soliloquy (loquor, Lat. 1 speak), a peaking alone, being the only speaker; called also a monologue; and in solifidian (fides, Lat. faith), one who supposes faith, and not works, alone necessary to justification. ** is the persuasion of the Solisidians, that all religion consists in onagarght.”—Hammond, **, of Baxon origin, from steopan, to bereave, whence the -o-o-on steep-cild, step-child, a child that is deprived of a * from this use the term steop or step was applied to rela- *d in a similar position, and thus we have steopop-mother; stoop-dohter, step-daughter; step-faeder, * *-sunn, step-son,

Sub, in Latin, under, as in subterranean (terra, Lat. the earth), under the earth; submersion (mergo, Lat. I dip), dipping; subscribe (scribo, Lat. I write), to write the name under a document. Suš may denote an inferior degree of the quality of the adjective to which it is prefixed, as sub-acid; sub-deacon, an under-deacon (diakonos, Gr. a servant). Sub becomes suc in succession, succeed, succinct, succumb, &c. Sof, in sufficient, suffuse, suffocate, suffragan, &c. Suy, in suggest, suggestion, &c. * — To nurse The growing seeds of wisdom that suggest, By ev'ry pleasing image they present, Reflections such as meliorate the heart, Compose the passions, and exalt the mind." Cowper, “Task."

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|tality of the Soul.”

Super, of Latin origin, the opposite of sub, signifies orer, above, as in supernatural, above nature; supermundane, above the world; supervision (video, Lat. I see), overlooking. “If a grammatical foundation be not laid deep at an early age, it will not often be laid in such a manner as to bear a large superstructure. -Anar. Sur, a French abbreviation of super, appears in surcharge, an orercharge, an additional charge; in surcoat, an overcoat; in surtout, literally an orerall (tout, Fr. all); in surfeit (faire, Fr. to do), an over-doing; that is, eating too much. “There are various degrees of strength in judgments, from the lowest surmise to notion, opinion, persuasion, and the highest assurance which we call certainty.”—Scarch, “Light of Nature.”

Syn, of Greek origin (syn, with), occurs in the forms syl, sym, syn; as in syllogism, symphonious, synchronous, &c. “Men have endeavoured to transforme logick, or the art of reasoning, into a sort of mechanism, and to teach boys to syllogise, or frame arguments and refute them, without any real inward knowledge of the question.”—Watts, “Logick.” “Up he rode, Follow’d with acclamation and the sound Symphonious often thousand harps that tuned Angelic harmonies.” Milton, “Paradise Lost.” “Sensations are impressed either at the same instant of time, or in contiguous successive instants. Hence it follows that the corresponding associations are either synchronous or successive."—Belsham, “Philosophy of the Mind.” Tetra, of Greek origin, signifying four, appears in tetragonal, four-angled; tetrameter, a line consisting of four measures or feet; and in tetrarch, properly a governor of a fourth part, a subordinate prince. “And Eroude tetrarck herde alle thingis that werem don of him."— Wiclis, “Testament,” Luke ix. 7. Trans, in Latin, across, as in transpose, to put across from one place to another; transport, to carry over the sea. “With transport views the airy rule his own, And swells on an imaginary throne." Pope. Tri, of Latin origin (tres, tres, tria, three), appears in triangle; trident (dens, Lat, a tooth), Neptune's sceptre; in trilateral (latus, Lat. a side), three-sided, and triliteral, having three letters, &c. “When a county is divided into three of these intermediate jurisdictions, they are called trithings. These trithings still subsist in the county of York, where, by an easy corruption, they are denominated ridings—the north, the east, and the west riding."—Blackstone, “Commentaries.” Pice, of Latin origin, signifying in the place of, as in ricegerent (gero, Lat. I bear), one governing as a substitute, viceroy, or “viceking,” see Hackluyt; also, vicechancellor, vicepresident.

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“In the yeare 1228, one Reginald was viceroy, or petie king of Mam."— Litvinshed.

Vicar (Lat. vicarius), comes from vice, and so denotes one who is in the place of another, hence a “vicarious sacrifice.” How strange that Richardson, in his Dictionary, should have given out the idea that vicar could come from viz (with difficulty), “one to perform a work of difficulty.”

“Nature, the vicare of the Almighty Lord, That hote, colde, hewie, light, moist, and drie Hath knit, by even number of accord, In easie voice, began to speak and say.” Chaucer. “Then it was devised that, by their common seal (which is the tongue of their corporation), they might appoint a deputy or vicar to do it for them.”—Spelman, “On Tythes."

Wiscount is made up of the same prefix; that is, vice; and comes. Lat. a companion, in low Latin count or earl; so that viscount (pronounced vi'count) is the deputy, the lieutenant of the count or earl.

“The viscont, called either procomes or vicecomes, in time past, governed in the countie under the earle, but now without any such service or office; it is also become a name of dignity next after the earle, and in degree before the baron.”—Holinshed, “Description of England.”

Ultra, of Latin origin (ultra, beyond), is used in ultramarine (mare, Lat. the sea), properly beyond the sea; applied to colour, fine blue. “Ultramarine or azure is a very light and a very sweet colour.”— Dryden, “On Painting.” The blue colouring matter of the lapis-lazuli, or azure stone. Vivi (Latin, vivus, alive) appears in vivify, to make alive; and in tiviparous (parie, Lat. I bring forth), bearing (its young) alive. “The usual distinction of animals, with respect to their manner of generation, has been into the oviparous (ovum, Lat. an egg) and viviparous kinds; or, in other words, into those that bring an egg, which is asterwards hatched into life; and those that bring forth their young alive and perfect."—Goldsmith, “Animated Nature.”

Un, of Saxon origin, not, reverses the meaning of the word to which it is prefixed, as unnatural, not natural, the opposite of natural. “Thus was I, sleeping, by a brother's hand, Of life, of crowne, and queene at once dispatcht; Cut off even in the blossomes of my sinne, Unhouzled, disappointed, unaneld.” Shakspeare, “Hamlet.” Unameld is unoiled, not having received the oil of extreme unction; disappointed means not prepared. To housel is to minister, the communion to one who is on his deathbed. Housel comes from the Saxon husel, the host, or sacrifice of “the sacrament of the Lord's supper.” Un, from the Latin unus, one, is exemplified in unanimous (animus, Lat. mind), of one mind; in uniparous, bearing one at a birth; in unison (sonus, Lat, sound), one single sound; in univocal (vox, Lat. a voice), having one voice or meaning. Un, of Saxon origin, has in some measure yielded to in; thus, for the old form unperfect we now say imperfect. “Unpossible" is quite obsolete. Under, of Saxon origin, is found in such words as undersell, underprop, undervalue, underwent. In the word understand, the derivative or secondary meaning is very remote from its primitive; namely, to stand under. Undertaker and underwriter have, in process of time, come to have very special significations. Undertaker, originally one who took on himself a certain duty, is at present applied to persons who are intrusted with the management of funerals; and underwriters, properly signifying those who wrote (their names) under a legal document, in Latin, subscriptor, is a word limited to persons who render themselves liable in a policy of marine insurance. Uni, of Latin origin (unus, one), occurs in unicorn (cornu, Lat. a horn), an animal with one horn; and uniform (forma, Lat. form), having one form. Up, of Saxon origin, is found in uphill, uphold, uplift, upspring. upstart, &c. Upbraid Richardson derives from a Saxon term not having the necessary import, and is in consequence obliged to twist his deductions into the most suitable shapes he can find. Upbraid comes from the classical Latin opprobrium, a reproach, through the low Latin word opprobrare, to reprouch. The up in this case comes from the Latin ob, changed for the sake of euphony into op aad up.

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Thus conjugate cupio 3, I wish, desire; facio 3, I do, makes fodio 3, I dig, jacio 3, 1 throw; pario 3, Ibring forth ; rapio 3, I plunder; sapio 3, I taste; &c. dico. 3, I say; duco. 3, I lead; facio, I do or make; fero 3, I bear; the 2nd person singular of the imperatives of these four verbs, are respectively die, duc, fac, fer. Let it be again remarked, that the participles in us are declined like adjectives in us, thus:– docturus, a, um docturi, ae, i docturo, ae, o docturum, am, um, &c.

amatus, a, tim amati, ae, i amato, ae, 0 amatum, am, um, &c. In all instances they must agree with their nouns. So also must the infinitive passive of the past tense, as eruditum esse, eruditam esse; eruditos esse, to have been instructed, the participle changing as the noun changes. The participle future in rus is frequently used after a verb, denoting motion to point out the object or design : *, Yoniunt expugnaturi urbem, they come with a view to capture the city. esides the conjugations now set forth, there is another recognised by grammarians. This is galled the Periphrastic conjugation. It is called periphrastic, (Greek, peri, about; and phrasis, a speech) because it is a kind of circumlocution; the thanges of idea not being expressed by additions to the stem, as in the ordinary conjugations, but by two separate words; Thus such a conjugation or form is made by the participles and the several tenses of the verb esse, to be , e.g.,

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future action.

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