perdu de nouveau. 19. Ce boulanger vous fournissait-il de bon pain? 20. Il nous en fournissait d'excellent. 21. Punissiez-vous souvent vos écoliers? 22. Je les punissais quand ils le méritaient. 23. Où étiez-vous ce matin quand je Vous cherchais? 24. J'étais dans ma chambre. 25. Je finissais mon thême.

EXERCISE 104.

1. Who was at your house this morning? 2. My friend G. was there, and was looking for you. 3. Did you speak to my father yesterday? 4. I was speaking to him when they brought me your letter. 5. Did your father used to wear a white hat when he lived in London? 6. He use to wear a black hat, and my brother wore a black coat. 7. Were you singing when my father came? 8. No, Sir, I was finishing my exercise. 9. Had you lost your pencil this morning? 10. I had lost it, and was looking for it when you spoke to me. 11. You used to like reading (la lecture), did your sister (use to) like it also? 12. She liked it also. 13. What song were you singing this morning? 14. I was singing an Italian song. 15. Have you been afraid to speak to me? 16. I have never been afraid to speak to you. 17. Have you brought my book? 18. I have not brought it.

SECTION LIII.

THE IMPERFECT (continued).

1. The imperfect of the indicative of every French verb, regular or irregular, ends in ais, ais, ait, ions, iez, avent. 2. No verb of the first conjugation, ER, is irregular in this

tense.

3. The only irregularity found in the irregular verbs of the second conjugation, IR, is that, to form the imperfect, the stem of these verbs takes ais, &c., instead of issais: as, ven-ir, je ven-ais; cour-ir, je cour-ais; cueill-ir, je cueill-ais. Exception: Fuir, to flee-je fuyais.

4. The irregular verbs of the third conjugation, orr, change that termination (oir) into ais, &c., like the regular verbs of the same; as, sav-oir, je sav-ais; av-oir, j'av-ais. Exceptions: se-oir, to become; voir, to see; and their compounds, and déchoir [see § 621.

5. The changes which the stem of the irregular verbs of the fourth conjugation undergoes, in this tense, are too various to admit of a complete classification. We, however, offer the following:

Sav-oir, 3 ir. to know.

Se tromp-er, 1. to be
mistaken.

Val oir, 3. ir. to be worth.

Ven-ir, 2. ir. to come, to
have just.
Vite, quickly.

15.

Parceque j'avais peur de me tromper. 3. Ne erai uiez-vous
1. Pourquoi n'écriviez-vous pas plus vite ce matin? 2.
pas d'offenser cette dame? 4. Je craignais de l'offenser, mais
je ne pouvais faire autrement? 5. Que peignez-vous ce
matin? 6. Je peignais un tableau d'histoire. 7. Votre tein-
turier que teignait-il? 8. Il teignait du drap, de la soie et de
la toile. 9. De quelle couleur les teignait-il? 10. Il teignait
le drap en noir, et la soie et la toile en vert. 11. Conduisiez-
12. Je conduisais mon fils ainé à l'église.
vous le jeune Polonais à l'école lorsque je vous ai rencontré?
13. Que lisiez-
vous? 14. Je lisais des livres que je venais d'acheter
Ne saviez-vous pas que ce monsieur est mort? 16. Je l'avais
oublié. 17. Combien la montre que vous avez cassée valait-
elle? 18. Elle valait au moins deux cents francs. 19. Ne
valait-il pas mieux rester ici que d'aller à la chasse? 20. Il
valait beaucoup mieux aller à l'école. 21. Votre ami que
vous disait-il? 22. Il me disait que son frère est revenu
d'Espagne. 23. N'alliez-vous pas à la chasse tous le jours
lorsque vous demeuriez à la campagne? 24. J'allais souvent
à la pêche. 25. Mon frère allait tous les jours à l'école quand
il était ici.
EXERCISE 106.

house? 2. I was afraid. 3. Of what were you afraid? 4. I 1. Were you afraid this morning when you came to our falling? (de tomber. See Sect. 20, R. 2. 4.) 6. He was not was afraid of the horse. 5. Was not your friend afraid of afraid of falling, but he was afraid of making a mistake (de ge tromper. See 2. in Exercise above). 7. Were you taking your son to school? 8. I was conducting him to school. J. What colour was the dyer dyeing the silk? 10. He was dyeing some red and some green. black or green? 11. Was he dyeing his linen cloth 12. He was neither dyeing it black nor green, he was dyeing it pink (rose). 13. What was the gentleman reading? 14. He was reading a leter which he had just received. 15. Were you cold when you came here? 16. I was cold, hungry, and thirsty. 17. Were you not ashamed of your conduct (conduite)? 18. I was ashamed of it. 19. Whither were you going when I met you? 20. I was going 6. Like prendre and écrire are conjugated, in this tense, to your house. 21. Were you driving your brother's carriage? those verbs in which prendre and crire appear in composition:22. I was driving my own (la mienne). 23. Were you writing as, comprendre, je comprenais; souscrire, je souscrivais.-Like to me or to my father? 21. I was writing to your friend's eraindre and connaître, those ending in indre and aître : teindre, cousin, je teignais; paraître, je paraissais.-Like conduire, those ending in ire: as, lire, je lisais; faire, je faisais ; luire, je luisais ; dire, je disais, &c.-Exceptions: rire, traire, écrire, and their compounds.

PRENDRE, to take.

ECRIRE, to write.

Je pren -ais, etc.

écriv -ais, etc.

Connaiss -ais, etc.

CONNAÎTRE, to know.

CRAINDRE, to fear.
craign -ais, etc.
CONDUIRE, to conduct.
Conduis -ais, etc.

7. Mettre and its compounds, and être are regular in this

tense.

LESSONS IN GEOLOGY.-No. XIV.
By THOMAS W. JENKYN, D.D., F.G.S., &c.

CHAPTER I.

SECTION IX.

8. The participle present, from which the French gram- ON THE ACTION OF VOLCANOES ON THE EARTH'S CRUST. marians derive the imperfect, presents of course the same irregularities: as, venant, valant, prenant, écrivant, craignant, connaissant, conduisant. Exceptions: avoir, ayant; savoir, sachant. RESUME OF EXAMPLES.

It had been long suspected that such volcanic phenomena were occasionally taking place in the bed of the sea. Officers and crews of vessels had frequently reported that, on their voyages, they had seen in different places sulphureous smoke, jets of flame, and spouts of water, rising up from the sea. coloured, and appeared in violent agitation, as if boiling. At other places the waters of the ocean were found greatly dissome points, shoals and reefs of rocks were observed as having just emerged, where, on a previous voyage, the water was known to have been many fathoms deep. These reports led scientific men to infer, that a power from below must be pro

pelling the bottom of the sea upwards towards the surface. This philosophical conjecture or inference has been established by a copious variety of facts, in the formation of new islands above the waters of the ocean.

The first well-ascertained instance of an island being elevated by a submarine volcano, was that near St. Michael in the Azores. In the same neighbourhood, various eruptions had been known in the years 1638, 1691, and 1719; but in the year 1811, a most terrific earthquake was felt at St. Michael. For more than half a year previously several shocks had been felt, but on January 31st, the convulsions were at the height of violence. On February 1st, at a spot about two miles out at sea, from the village of Genites, volumes of sulphureous vapours were seen to rise out of the sea, which spread themselves in all directions. These were accompanied with jets of fire. At the same time, the wind carried volcanic ashes from the sea as far as the town of Ponta del Gada, 18 miles off, where they fell and covered the houses and the adjacent fields. The columns of ashes and erupted materials, as they were rising from the sea, could be seen for many miles round, and appeared by night like pillars of fire. While these were rising, the sea boiled in terrible agitation.

In about eight days these eruptions ended, and the bottom of the sea was raised nearly to a level with the water. This was in a part of the sea which was known to be from 300 to 500 feet deep. On June 13th another earthquake announced the approach of another eruption, which broke out at the distance of two miles and a half from the other spot, a little to the west of Cape Pico das Camarinhas, and on the 17th it was at its greatest violence. Columns upon columns of ashes and smoke rose at intervals with fearful agitations, to the height of many hundred feet above the sea, and spread themselves out in thick clouds, which were rendered more terrible by frequent flashes of lightning.

In 1831, at a spot about thirty miles to the south-west of Sicily, a submarine volcano rose out of the sea, and formed an island. Before its appearance, it was well known that the depth of the sea at that place was 600 feet. The process of its rise was this. First, there were violent spoutings of steam and water from the bed of the sea, jetting sixty feet high. Then a small island of dry ground appeared with a burning crater in the centre of it. This crater ejected ashes, scoriæ, and thick volumes of smoke; and the whole sea around became covered with floating cinders and with shoals of dead fishes.

This volcanic island rose gradually till it reached the elevation of nearly 200 feet, with a circumference of about three miles at the base. In its centre was the crater, which was now a basin, six hundred feet in diameter, full of dingy red water in a boiling state, and continued so for three weeks. This volcanic island continued above the sea for nearly three months, and then it sank gradually again into the sea. Before it began to sink, its circumference became much diminished by the continued action of the waves on all its sides. It appeared July 18, 1831. Towards the close of October, the whole was nearly on a level with the surface of the sea. After it disaplig. 30.

At the close of this eruption, an island became visible, and rose gradually to the height of three hundred feet. It had, at one end, a summit, in the form of a cone, and at the other, a deep crater, out of which violent flames of fire were gushing, though it was under water at full tide.

Captain Tillard, who was in the neighbourhood, visited the island, and called it, after the name of his ship, "Sabrina." He found its mass of ashes and cinders too hot for walking on it. He could see that when the tide returned, the sea flowed with tremendous violence into the burning crater, where the water was boiling as in a hot caldron. Through the continued eruptions of burning stones, sands, and ashes, from the crater, the conical hill already mentioned, on one side of the island, rose eventually six hundred feet above the sea. After all, in the last days of February, 1812, the entire island'sank into the sea and completely disappeared. The annexed engraving (fig. 30) will assist you in imagining the appearance of these phenomena at sea.

In a former lesson it was mentioned that some of the cavities of subterranean fires must be of immense areas. This is proved by the fact that the reservoir of volcanic fires, which lies under the southern part of Italy, extends far and wide beneath the bed of the Mediterranean, and sometimes occasions the rise of fresh shoals and new islands in that sea.

peared, it left behind it a dangerous reef of hard volcanic rock just eleven feet under water, encompassed by shoals consisting of scoriæ and sand. Fig. 31 is a representation of its general appearance when at its highest elevation.

This little island received seven different names. It was well known by the name of Ferdinandea; but the name "Graham Island" has been fixed upon by both the Royal and the Geographical Societies.

I want you now to apply your geological knowledge to the investigation of this phenomenon. Here was a sea 600 feet deep, and here is an island raised, in a few weeks, 200 feet high. Here is, therefore, a quantity of land three miles in circumference, raised up to a

total elevation of 800 feet in a very brief period. The shoals of dead fish, which were found in the volcanic sands around it, will explain similar facts which have been discovered in strata connected with volcanic districts. When the crater of this little island was ejecting mineral masses, it is probable that they would envelop some of the dead fish at the sea bottom, and that, when the erupted ashes fell again, both they and the fish which they contained were ingulfed in the bottom of the ocean. You can imagine that if ever this sea bottom will become elevated above the waters of the sea, and be explored by some future geologist or ichthyologist, the fossil fish of the Mediterranean, imbedded in volcanic tufa, will prove an important study.

Graham Island presents to you the advantage of having been carefully examined by scientific men. On this account, the study of its structure will much aid your inquiries into the volcanic formation of rocks. As much of the island as was visible was formed of loose incoherent materials, such as sand, scoriæ, pumice, &c., ejected from the crater. These loose materials, after having been hurled to a considerable height, fell again on each side of the central basin, and settled down in regular strata, as represented by the dark lines on the left and the right of tig. 31, and parallel to the deep inward slopes of the crater. But at some distance from the rims around the

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 pheno-Wo mena of submarine volcanoes. I would, otherwise, have referred you to the rise of an island called Nyöe, on the coast of Iceland, during the tremendous eruption of Skaptar Jokul, which occurred in 1783; an island which afterwards disappeared: and to another which, in the form of a peak, with some low

andern bleiben in New-Yerk. 18. Die Einwanderer in Amerika find Auswanderer aus Europa und aus andern Theilen der alten Welt. 19. Wann wollen Sie auf das Feld gehen? 20. Ich bin schon auf dem Felde gewesen, und kann nicht wieder dahin gehen, aber ich muß jezt bald in den Garten gehen, denn mein Lehrer ist da und will mich sehen. 21. Warum will tiefer Italiener nicht englisch sprechen? 22. Er wollte es wohl (Sect. 44. IV.) sprechen, aber er kann es noch nicht; er spricht nur ita. lienisch und spanisch. 23. Wie vil Sprachen können Sie sprechen? 24.

1. Wo ist ter Schwager? 2. Er ist an dem (am) Tische. 3. Wo geht | Ich spreche nur zwei, aber ich will noch andere lernen. der Zuckerbäcker hin? 4. Er geht in tie Backstube. 5. Wo ist sein Freund, der Schauspieler? 6. Er ist in dem Opernhause. 7. Wo geht sein Freund, der Seiler, hin? 8. Er geht in seine Werkstatt. 9. Wo ist der Hirte? 10. Er ist auf dem Verge. 11. Wo geht der Hirte hin? 12. Er geht auf den Berg. 13. Wo geht unser alter Nachbar hin? or, Wehin geht unser alter Nachbar? ($ 89. I.) 14. Er ist jest in dem kleinen Garten, aber er geht bald in den großen Garten. 15. Seine Frau ist in diesem Hause, aber sein Vetter geht in jene Viltergallerie 16. Ich stehe an tem (am) Fenster, und sie kommen aus (§ 4. 2) Fenster. 17. Der Ritter fist schen auf seinem guten Pfeite, und der Kuecht springt auch so eben auf sein gutes Pferd. 18. Der Mann sigt am (§ 4. 2) Tische, und das Buch liegt auf dem Tische. 19. Ich habe keinen Hut auf dem Kepfe. 20. Wo geht ter Seltat hin? 21. Die Soltaten gehen aufs (§ 4. 2) Feld; fie find schon auf dem Felte. 22. Der Fresch springt in den Fluß und schwimmt in dem Flusse, und tie Gans fthwimmt in tem Teiche. 23. Ich habe diese Werte irgendwo gelesen. 24. Ich kann meine Müge nirgents finden, obgleich fie irgentwo in diesem Zimmer sein muß.

1. When did he live? 2. He lived in the fourteenth century1. 3. My friend told me he would never go there again?. 4. Do mander-in-chief has sent his troops where the danger was most3. you go to Spain? 5. No, I shall not go thither. 6. The com7. Is this ship from Spain or from Havre? 8. No, it is neither1 from Spain nor from Havre; it comes from Hamburg. 9. These immigrants are going to Milwaukee, and are emigrants from Bohemia and Venedig. 10. Can you leap over that gate? 11. I could when I was young. 12. He hade me to go thither that he might speak to me about it.

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 revenge3 and whetted sword of a traitor enter not2;-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.

2.

3.

1. Die Soltaten find hier, und der Feltherr kommt auch hierher. Der Feind ist schon da, und unsere tapfern Brüder müffen dahin ziehen. Wann gehen sie nach Spanien? 4 Ich will gar (Sect. 15. III.) nicht tahin gehen, aber mein Vater will in nächster Woche dahin reisen. 5. Sind Sie schon da gewesen? 6. Nein, aber einer meiner Bekannten war da und will nie wieter dahin gehen 7. Wir gehen auf ten Berg, wellen Sie mit uns gehen? 8. Will ter Russe seinen Bedienten in die Start schicken? 9. Er hat ihn schen rahin geschickt. 10. Werten die Truppen hierher kom men? 11. Sie werten nicht hierher kommen. 12. Wo kommen diese Fremten her? 13. Sie sind Einwanderer und kommen aus Böhmen. 14. Ist dieses Schiff von Bremen oter Havre? 15. Es ist weter von Bremen, noch von Havre, es ist von Venetig. 16. Gehen diese französischen Eins wanterer nach Milwaukee? 17. Ein Theil von ihnen geht dahin, tie

lim vierzehnten Jahrhundert. wieder. die meiste Gefahr, weter. noch. Thor. 7bat.

LESSONS IN GEOMETRY.-No, XI.

LECTURES ON EUCLID.

DEFINITIONS. BOOK I. FROM XIII. TO XVIII. INCLUSIVE.
XIII.

[A term or boundary is the extremity of any thing.] This definition is unnecessary, being merely verbal.

XIV.

A figure is that which is enclosed by one or more boundaries. This defi 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.

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 mind of what are called geometrical figures, which are entirely every kind; but it does not apply to the ideas formed in the 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, thereticulars of importance relating to the circle. We find that fore, it will only be necessary to advert to some additional pargeometers 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 exdefined in the 15th definition, is meant the space contained clusive of the space contained within it. Now by the circle, as within the boundary, and the boundary too; for we cannot conceive of space of a certain form without including its out,

line 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 inclled within the arc and its chord is called a segment, i. e., a cating, 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,

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 100% or 15; 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 ; 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. Original Ratios. 7:22 106: 333 1250: 3927

Authors.
Archimedes
Lambert

Hindoos
Metius
Lambert
Lambert
Lambert
Wallis

1: 2

1416.

113: 355
33102: 103993
33215: 104348
66317: 208341

22 42 69 82
3.5

The Same in Decimals.
1: 3.142857142857-
1: 3141509433962+
1: 3-141600000000
1: 3.141592920353+
1: 3.141592653011+
1: 3.141592653921+
1: 3141592653467+

67&c.) 1: 3141592653589+

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 diameter 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 connoisscur could ever reach; so in the mathematical sciences, there is an accuracy of ratio or proportion, which the finest infinity, the numerators of the fractions in the parenthesis being struments ever constructed, or ever to be constructed by 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 expressed in the ordinary terms of the decimal notation. It is between 38 and 3 as found by Archimedes; that is, the circumference of a circle is less than 348 times the diameter and greater than 3 times the diameter. As 348 is the same as 34 or 2, it is usually said that the ratio 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 34 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.

man, can never attain.

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

In the latter ratio, that of Wallis, the series continues to inthe 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 being down the diameter is 4 millions. This is what we call " upon the eircle" but we must delay the rest of our lesson on this subject till our next opportunity.

LESSONS IN ENGLISH.-No. XVI.
By JOHN R. Beard, D.D.
PREFIXES (concluded).

Quadr, quadra, of Latin origin (quatuor, four), is found in quadrangle, four-angled; quadruped (pes, Lat. a foot), four-footed; quadruple (plica, Lat. a fold), fourfold; also quater, as in quaternion (quaternio, the number four), &c.

"Aire and ye clements, the eldest birth

Of Nature's womb, that in quaternion run,
Perpetual circle, multiform; and mix

And nourish all things." Milton, "Paradise Lost."