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black paint, and after drawing a perpendicular line, A A, and a lens, and the same rays are now scattered outward, and be horizontal one, which will be the diameter of the circle, imagine come divergent. the horizontal line to represent the surface of the water, and the Lenses have varios figuros, and the lines that bound them other will of course be perpendicular to it. Let B C be the incident may be portions of circles or ellipses, or they may be right ray, and c d the refracted ray; if a dotted line is drawn from lines. Generally speaking, one or both sides aro portions of a the extremity of the arc at B to the diameter A, that will be the spherical surface, or one side may be s portion of a sphere, and sine of the angle of incidence, and the other dotted line, drawn the other a plane surface. Thus there are plano-conver or from the extremity of the arc at D to the diameter A, will be plano-concave lenses, one side of which would be flat, and the the sine of the angle of refraction; then if the sine of the angle other curved; or concavo-convex lenses, concave on one side of incidence, which may be supposed to be four inches, is and convex on the other, and if the concavity exceeds the condivided by the sine of the angle of refraction, ascertained by vexity, it would be regarded as a concave lens belonging to the direct experiment to be three inches, the quotient will be divergent class. A most useful lens is the meniscus, meaning 1.333, which is called the index of refraction for water. If the a little moon or crescent, one of whose surfacos is convex and lines are drawn upon the plate with black varnish, the plate the other concave ; but as the convexity exceeds the concavity, can be held upright in water, and the young student may trace this would be considered to be a convex lens, and must be out and study more practically that which might otherwise classed with the convergent lenses. puzzle him.

Before there were such facilities for obtaining glass lenses of There are particular positions, as when light passes from a almost any size or shape, it was always thought necessary to dense into a rare medium---viz., from glass into air—in which give a description of the mode of grinding and polishing lenses the refracted ray becomes parallel with the surfaces of the glass in works on optics; and as many of our younger readers may and air. At a more oblique angle, when light passes throngh have lathes, and would like to be able to say that they had the denser medium, and becomes incident upon the surfaces of constructed a simple telescope, and ground their own lenses, the glass and air, the ray is no longer refracted, but undergoes the following particulars, published many years ago, will be total reflection. This fact is well illustrated by placing an found to be eminently practical, and capable of giving fair engraving, E E, behind a prism, P, placed as in Fig. 5.

results The prism and picture should face the window, and if they “Manner of Grinding Lenses.-A little piece of copper is are placed on a stand level with the eyes, it is curious to notice, cemented to the end of the arbor of a lathe, and turned till as the spectator walks round, that there are certain vory it forms a dish or bason of the diameter of the lens required. oblique positions at the sides, AA, where the light from the Then a piece of clear glass is cemented, on one side of its window only is reflected, i nd no picture is visible, and where flat sides, to the end of a little marurel, with black Spanish the reflecting surface shinys like silver, because total reflection wax; and thus ground, on the side not cemented, on a grind. occurs; but as the observe: moves round in a half circle, say stone with water, till it hath nearly acquired a convex úgure. from 5 to B, the picture reappears, and again disappears as he It is finished in the lathe by turning it in the basin with fine passes to the opposite sid, and looks very obliquely at the wet sand, grit stone, or emery. The grit must be often repeated surface behind which the picture is placed. The brilliancy of fresh till the lens appear very round; when it comes to that the diamond is greatly owing to the total reflection of light, point they cease to take any fresh grit stone, but continue to which becomes visible at smaller angles of incidence in conse- turn it in the basin, till the remains of the sand are become 60 quence of the very high refractive power of this, the purest fine as to have polished it. This they perceive when, upon natural form of carbon.

wiping it, the image of the window of the place is seen A lens, in dioptrics (from olontpov, a perspective instrument), painted on its surface; if it does not, it is rubbed in water is defined by an old author to signify a small roandish glass of without any sand, and turned till it hath got a polish. The the figure of a lentil, which, in scientific botany, is called the basin is then covered, withinside, with two or three folds of lens; and this Latin word is said to have originated from lenis linen, and the polish finished with putty powder, or tripoli of (mild), because those who fed upon this sort of pulse were sup- Venice steeped in water. It is known to be perfectly polished posed to become mild and gentle in disposition. There are two when, viewing it with a magnifier, there appear no soratches great classes of lenses, called convergent and divergent lenses. of the sand. The cement is then broken off, and the side If the properties of concave and convex mirrors are understood, polished cemented, to work and grind the other, as before, till it is easy to remember those of lenses of the like figure, because the edges of the lens become sharp, and it be perfectly polished the latter have exactly opposite properties to the former. on either side. When finished, it is washed in spirits of wine,

A double convex lens is a good example of the convergent to take off all remains of the wax." class : divergent rays become parallel if passed through a lens According to the mode now generally practised, optical of this shape, and parallel rays are made so convergent that they glasses are fixed on blocks by means of a cement, and ground meet at a point called the focus, and termed the principal focus. with emery on a tool of proper convexity or concavity ; if they The rays of light from the sun are nearly parallel, and hence are small, a large number is fixed on the blocks at the same " the principal focus," or fire-place, of a double convex lens time. The tool is sometimes first turned round its axis by corresponds with that spot where the greatest heat is accumu- machinery, and when the lenses are to be finished, a compound lated, as in a burning glass, so that a double convex lens is a motion is given to it by means of a crank; and in order to simple form cf burning glass.

make it work smooth, the wheels turn each other by brushes In the illustration (Fig. 6) the divergent rays passing from an instead of cogs. The point of the lens where its two surfaces apertaro, E, in the copper chimney of an argand oil or gas light, are parallel is determined by looking through it at a minute fall upon a double convex lens, A A, and by refraction become object while it is fixed in a wheel with a tubular axis, and shiftparallel, and fall' upon the screen of paper, s; by reversing the ing it until the object appears no longer to move; & circle is description, and starting from s, as the source of parallel rays, then described as it revolves, in order to mark its outline. they are collected by An, and meet at the foont, or tre-place, E. The dishes in which lenses are ground are of bell metal, and

A double concave lens (Fig. 7) is a good example of the the emery is prepared by elutriation. The writer has seen fire divergent class. Rays of light already divergent become still hundred spectacle-glasses ground and polished at the same more so if allowed to fall upon a lens of this form. Parallel time by machinery at Sheffield, the operation being principally rays are made divergent, and even convergent rays are turned conducted by women, who exhibited the greatest dexterity in all in the opposite direction, and made less so by a double con- the manipulations, such as cementing the glasses on to the cave lens.

tool, and adjusting the basins and emery-powder to the work With two prisms the principle of the doable convex or con- required to be done. cave lens is demonstrated in the most instructive manner. By The focal length of any convex lens is easily found, by holdplacing the prisms base to base, and passing a pencil of sun- ing its axis in a line with the sun; the burning-point, or the light from a hole in a shutter through them, the rays are bent place where the rays are concentrated to the smallest speck, is inwards, and converge to a point, as they would do with a its focus; the distance of that focus from the lens is its focal convex lens, and this is easily seen by referring to Fig. 8. length.

When the position is reversed, as in Fig. 9, and the prisms The nearer an object is brought to the focus of a conter lors - held edge to edge, they virtually form a double concave the larger will be the image. The brightness of an imars

Or

increases with the size of a convex lens, but decreases in sharp- Ilava, I make tol police, policy, Ilope vw, I bring, ness; for only the rays that fall in the central part of the lens cease; in the mid

politic, political, bring forward; in converge to a point; those towards the edge disperse as in a dle, I cease polity), a state, the middle, I go, prism, and make the image of the object confused. Hence the

stop.

constitution, go- proceed, travel, lens of the eye is covered with the iris, except at its centre; TOÀiteia, -as, n (from vernment.

Ilu11, -775, i, a door, and reading glasses have their edges ground off or covered with TOIS : hence our

gate. black horn. This defect is called the "aberration of sphericity." It is, perhaps, difficult to say who made the first telescope.

EXERCISE 90.-GREEK-ENGLISH. The conception of such an optical instrument appears to belong

1. Ο! πολεμιοε επι την ημετεραν πολιν στρατεύονται. 2. Ιερε to Friar Bacon, as it is asserted by Dr. Jebb, who edited the της των πολιτων σωτηριας βουλευσομεθα. 3. Ο πατηρ μοι ελεγεν famous work of Roger Bacon entitled “ Opus Majas,” that in ότι πορευσοιτο. 4. Οι Ελληνες επι τους Περσας εστρατευσαντο. one of the passages of this work the friar states he actually 5. Αναπαυσωμεθα, ω φιλοι. 6. Προ του εργου ευ βουλευσαι. 7. applied telescopes to astronomical purposes, and therefore so Παντες τιμης γευσασθαι βουλoνται. 8. Ο πατηρ αναπαυσαμ long ago as the thirteenth century. Afterwards the names of TOPEVOETUL. 9. Αί πυλαι της νυκτος κεκλεισoνται. , 10. Eay Baptista Porta, Digges, and then Jansen and Galileo, were τοιουτος ανηρ την πολιτειαν επιτηδευη, αυτη ευ βεβουλευσεται. connected with this important instrument.

REMARKS ON THIS EXERCISE. A very simple astronomical telescope can be made with two

FlopevoolTo, the optative, because it is preceded by an historical paper, wooden, or brass tubes, sliding one within the other. At tense, and because the action depends on the words or declaraone end of the widest tube is fixed the object-glass, a double tion of the subject matnp (obliqua oratio). convex lens with a long focal distance, and called the objectglass because it is nearest the object; while the lens placed at The force of the participle in Greek can often be given in Eng

Avanavgalevos, having rested; that is, when he has rested. the other end of the smaller tube, through which the observer lish only with the aid of a conjunction or an adverb. looks, is called the eye-glass, also a double convex lens, but having a short focal distance. An inverted image is formed in

TYS VUKTOS, by night, the genitive of time. (See the Syntax.) the focus of the object-glass, and this is magnified again by

EXERCISE 91.-ENGLISH-GREEK. the eye-glass. With this telescope all objects are inverted, 1. I shall have been educated. 2. He will have been educated. and therefore the one invented by Galileo (Fig. 10) is the more 3. We shall have been educated. 4. They will have been planted. convenient. It is constructed like the astronomical tele- 5. He will have been slain. 6. The general will march to the scope, only a double concave lens is substituted for the double city. 7. The general marched to the city. 8. The general may convex one used as the eye-glass. It is, in fact, similar to an have marched to the city. 9. The general might have marched opera-glass, and has this advantage, that the object is seen to the city. 10. We shall have consulted respecting the safety erect.

of our native land. 11. He will consult respecting thy safety. Mr. Richard A. Proctor, in his excellent work entitled “Half 12. He consulted respecting the safety of the citizens. 13. Hours with the Telescope,” says: “ There are few instruments They ceased. 14. They will have ceased. 15. He will cease. which yield more pleasure or instruction than the telescope. 16. The two men ceased. 17. We will cease, O friends. 18. Even a small telescope-only an inch and a half or two inches The friends travel. 19. The friends will travel. 20. The friends perhaps in aperture-will seem to supply profitable amusement travelled. to those who know how to apply its powers. I have often seen with pleasure the surprise with which the performance of an

THE FIRST AORIST AND THE FIRST FUTURE PASSIVE. opera-glass, well steadied, and directed towards certain parts

The first aorist passive is formed from the stem of the perfect of the heavens, has been witnessed by those who have sup- active by changing into @nu, and by changing the reduplication posed that nothing but an expensive and colossal telescope into the syllabic augment, as AcAur, cauony. could afford any views of interest. But a well-constructed

The first future passive is formed from the first aorist passive achromatic of two or three inches in aperture will not merely by dropping the augment and changing v into aouai, as edunv, supply amusement or instruction--it may be made to do nsefal Aventojai. work." The principles of the achromatic telescope will be

VOCABULARY. considered hereafter.

Δημοκρατια, -ας, ή, bellum infero), to treaty; in the text, democracy, the make war on.

used in the plural,

government of the Mn, not, last (Latin, the agreementsLESSONS IN GREEK.-XXXI.

onuos or people ne).

that is, the treaty

(that is, the popu- Toneulos,-c, -ov, hos- considered as conTHE FUTURE AND FIRST AORIST MIDDLE AND THE PERFECT

lace).

tile, the enemy's. taining many

Επιφερω, I bring up- Συνθηκη (συν 2nd heads. The future middle is formed from the future active by chang- on, I introduce ; τιθημι), a conven- Τυραννος, -ου, ο, και ing the personal ending of the active-that is, w-into the per- TTONELOV TIVI (Lat. tion, agreement, tyrant. sonal ending of the middle---that is, ouau: as avo-w, Avo-opas. The o here may be considered as a connecting vowel, and the

EXERCISC 92.GREEK-ENGLISH. mood be divided thus--Au-o-o-jar. Of each of these four parts 1. Εκτωρ υπο Αχιλλεως εφονευθη. 2. Τω αδελφω υπο του αυτου the student should give an account.

διδασκαλου επαιδευθητην. 3. Πολλαι δημοκρατιαι υπο των τυρανThe first aorist middle is formed from the future middle by νων κατελύθησαν. 4. Μεγας φοβος τους πολιτας έχει, μη αι συνprefixing the augment and changing oμαι into αμην-thus, Θηκαι υπο των πολεμιων λυθωσιν. 5. Ειθε παντες νεανιαι καλως λυσ-ομαι, ε-λυσ-αμην; Or it may be formed from the first aorist | παιδευθειεν. 6. Φονευθητι, ω κακουργε.

7. Οι στρατιωται €15 active by simply adding μην--thus, ελυσα, ελυσα-μην

την πολεμιαν γην πορευθηναι λεγονται. 8. Οι πολεμιοι, των συνThe perfect future, or, as it is sometimes called, the third θηκων λυθεισων, ημιν πολεμον επιφερουσιν. 9. Ο ληστης φονειfature (also the paulo-post-futurum), is formed from the second ondetai, person singular of the perfect passive by changing au into oual,

REMARKS ON THIS EXERCISE. as AeAupai, Aelvo-quan-where, again, o may be accounted a Tupavyos does not exactly correspond with our word tyrant, connecting vowel as well as the model vowel, or vowel marking though the latter comes from the former, but denotes one who the indicative mood. For the optative, o becomes ol, as AeAvool- has seized the helm of government in a free state. A tyrant, ana_that is, « is added to o.

therefore, in the Greek sense of the term, is not necessarily a The principal parts of wavw are, Tavw, Tavow, TTETRUKA, TETAUO- despot, and the Greek may often be rendered by our usurper. pas; the future middle, ravoquai; first aorist middle, eravo aumu; Mn after verbs expressive of fear may be rendered by lest, perfect future, πεπαυσομαι.

and requires the subjunctive with a present, a perfect, or 2,

future tense ; and is followed by the optative when the verb in VOCABULARY.

the principal sentence is in an historic tense. Averava, I cause to rew, I let taste ; in Etindeuw, I attend Evvonkwy AvbeltWY. This is what is called “the genitive

rest; in the mid- the middle, I taste to, I prosecute, absolute," and corresponds with “the ablative absolute" i dle, I rest. (with genitive). practise.

Latin-the treaty being broken.

FUTURE.

B

E

EXERCISE 93.- ENGLISH-GREEK.

EXERCISES IN EUCLID.-V. 1. They will be slain. 2. They were slain. 3. He was slain. 4. Two soldiers were slain. 5. Many men will be slain. 6. I

PROPOSITION XXIX.-- To trisect a given right angle; that is, shall be educated. 7. He will be educated. 8. We shall be to divide it into three equal parts. educated. 9. Ye two will be educated. 10. I was well edu- Let BAC (Fig. 28) be the given right angle; in ac take any cated. 11. The constitution was destroyed. 12. The constitu. point D, and on ad describe an equilateral triangle ADE; tion will be destroyed. 13. The treaty was broken. 14. The bisect the angle EAD by the line A F,

E
treaty will be broken. 15. The treaty being broken, the citizens meet ED in F; then the lines A E, AF
were slain. 16. The robbers were slain. 17. The robbers are will trisect the given right angle. For
said to have been slain. 18. The democracy will be destroyed.

since the three angles of a triangle are
together equal to two right angles, and

the angles of an equilateral triangle are KEY TO EXERCISES IN LESSONS IN GREEK.--XXX.

equal, the angle E AD is equal to oneEXERCISE 84.-GREEK-ENGLISH.

third of two right angles, i.e., to two1. The soldiers have slain two thousand two hundred and sixty-five thirds of one right angle. But the whole

Fig. 28. of the enemy. 2. Pherecydes used to say that he had sacrificed to no BAC is a right angle; therefore the regod. 3. As you are (having been produced) young, learn many good mainder B A E is equal to one-third of a things. 4. The soothsayer has foretold the future well. 5. You have right angle. And the angles E A F, FAD are each half the angle instructed your children well. 6. Medea, having slain her children, EAD; therefore each of them is one-third of a right angle; rejoiced. 7. The Lacedæmonians had destroyed Platæa. 8. Sardana. hence A E, A F trisect the right angle. Q. E. F palus had put on a woman's garment. 9. When the sun had set, the

PROPOSITION XXX.-If two right-angled triangles have one enemy approached. 10. Alexander, in his pursuit of (pursuing)

Darius, side and the base in the one equal to one side and the base in the king of the Persians, had made himself master of great wealth.

the other, each to EXERCISE 85.-ENGLISH-GREEK.

each, they shall be 1. Πεφονενκα. . 2. Πεφονευκασιν. . 3. Επεφονευκει. . 4. Φονεύσουσιν.

5.

equal in every respect εφονευσεν. 6. Φονεύσομεν. 7. Πεφονευκαμεν. 8. Επεφονένκειμεν. 9. Θυ

Let ABC, DET σουσιν. 10. Τεθυκασιν. 11. Ετεθυκεσαν. 12. Eθυσαν, 13. Ο μαντας τη θεω

(Fig. 29) be two triεθυσεν. 14. Ο μαντις τω θεώ βους εκατον τεθυκεν. 15. Παιδευω τα τεκνα. 16.

angles, having ABC, Επαιδευον τα τεκνα. . 17. Παιδευσω τα τεκνα. . 18. Επαιδευσα τα τεκνα.

.
19.

DEF right angles, and λεπαιδευκα τα τεκνα. 20. Επεπαιδευκειν τα τεκνα, 21. Αλεξανδρος Βαβυλωνα

let AB, AC in the one κατελνσεν. . 22. Αλεξανδρος Βαβυλωνα κατελελυκει. . 23, «Ο παις στολην

be equal to DE, DI γυναικειαν ενδυει. 24. Ο παις στολην γυναικειαν ενδεδυκεν. 25. Ο παις στολης

in the other; then γυναικειαν ενεδεδυκει. 26. Ο παις στολην γυναικειαν ενδυσει.

shall bc be equal to

с EXERCISE 86.-GREEK-ENGLISH.

Fig. 29.

E F, and the triangles 1. Two men are fighting. 2. Let us fight bravely for our country.

equal in every re3. It is necessary that a son should obey his father. 4. Many good spect. For if Bc be not equal to EF, one of them must bə men are poor. 5. It is honourable to obey the laws of the country. greater. Let Er be the greater, and from EF cut off EG= 6. Do not welcome those of your friends who gratify you in bad things. BC; join Do; then since A B, BC are equal to DE, EG, each to 7. Let each go quietly along the middle of the road. 8. Let the citizens each, and the included right angles are equal, therefore base ac obey the laws. 9. My two brothers follow me. 10. If you are willing is equal to base DG (Euc. I. 4). But Ac=DF by construction; to do well, work. 11. If you wish* (should you wish) to do well, work. therefore DG = D F, and angle DGF= angle DFG (Euc. 1. 5). 12. No one who lies is concealed for a long time (i.e., no one lies for a long But since D EG is a right angle, DG E is less than a right angle time without being found out). 13. The Lacedæmonians used to go on (Euc. I. 17); therefore D G F, which with DG E is equal to two consult without anger. 15. Two beautiful horses were driven into the right angles (Euc. I. 13), is greater than a right angle. Hence sity. 16. If you are poor you have few friends.

DG F and D FG together are greater than two right angles,
which is impossible by

A
EXERCISE 87.-ENGLISH-GREEK.

Euc. I. 17: hence EG is
1. Εκεινος πενεται και ολιγους φιλους εχει. 2. Πενομαι. 3. Eβουλευετο. 4. | not equal to BC. And
Εβουλευεσθην. 5. Βουλευομαι. 6. Βουλευεται. 7. Bovdet kalwr #PATTEIN, similarly it may be proved
εργαζου. 8 Εαν βουλη καλως πραττειν εργαζου. 9. Εργαζονται. 10. Καλως | that no line but ΕFis equal
εργαζεται. 11, Ειργαζετο. 12. Ειργαζεσθε. 13. Ειργαζεσθον. 14. Εργαζο- | to BC; hence EF is equal
μεθα. 15. Μαχομαι. 16. Εμαχομην. 17. Εμαχοντο. 18. Mαχονται γενναίως. | to Bc, and the triangles
19. Mαχεσθε. 20. Εμαχεσθε. 21. Ω στρατιωται γενναιως μαχεσθε περι της
πατριδος. 22. Καλον εστι περι της πατριδος μαχεσθαι. 23, Σοι έπομαι. 24.

equal in every respect.

B

CE Εμοι έπεται. 25. Εμοι έπονται, 26. Τη στρατηγω επομεθα, 27. Τη στρα

Q. E. D.

Fig. 30. νευματι είπομεθα, 23. Τοις νομοις, ω παιδες, έπεσθε.

Corollary. This propo

sition is not necessarily true, as might be supposed, if the equal EXERCISE 88.-GREEK-ENGLISH.

angle in the two triangles be not a right angle; for in this case 1. The robbers have been slain. 2. Two brothers have been edu- it is not necessarily untrue that DG is equal to DF. If the eated by the same master. 3. The monarchy has been destroyed by equal angle be not a right angle, there will be two positions the people. 4. Many temples to the gods have been built by the possible for the third side, as in Fig. 30, DF and D P' both Athenians. 5. Let the door be shut at once. sonsulted well before acting (lit., before the deed)." 7. The desire of sexe satisfying the conditions of the proposition. This is, of course, government is implanted in all men. 8. Let the robbers be slain at

the "ambiguous case" of trigo once. 9. The enemy aro said to have been shut up in the citadel. 10.

nometry. Lenophon's two sons, Gryllus and Diodorus, had been educated in

PROPOSITION XXXI.-The straight Sparta.

lines which bisect the angles of EXERCISE 89,-ENGLISH-GREEK

triangle meet in a point. 1. Πεφονενται. 2. Οι παιδες πεφονεύνται. 3. οι στρατιωται επεφονέυντο.

Bisect the angles ABC, BCA 4. Κατακεκλεισται. 5. Κατακεκλεισθε. 6. Κατεκεκλεισθε. 7. Κατακεκλεισ

c (Fig. 31) of the triangle ABC by 8. Ω δυο ανθρωπω κατεκεκλεισθην. 9. Οι βους κατακεκλεισθαι

Fig. 31.

the lines BG, CG meeting in G Χεγονται. 10. Εν πεπαιδευμαι. 11. Εν επεπαιδευτο, 12. Εν πεπαιδευνται.

join Ag; then shall AG bisect the 13. Κακως επεπαιδευμην. 14. Κακως επεπαιδευσο. 15. Τα δενδρα εν πεφυτευται. | angle BAC. Draw α D, G E, GF perpendicular to A B, BC, CA 16. Τα δενδρα κακως έπεφυτευτο. .

then in the two triangles GDB, GEB, since the right angle

GDB is equal to the right angle G E B, and by construction the The difference between a Bovla and cav BovAy may be thus ex: angle a BE is equal to the angle & BD, and the side G B is complained: a Boulet assumes that you are willing - If you are willing, which mon; therefore by Euc. I. 26 the triangles are equal, and thereI believe you to be-and so may be translated since ; eav BovAy makes no fore GD= G E. By an exactly similar course of reasoning (8 such assumption-should you be willing, about which I express' no opinion =G F, therefore GD=GF; and because GD=GF, and G A is sither way.

common, also right angle a DA= right angle a FA, therefore,

D

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B

E

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G

B

E

by the last proposition, the triangles are equal; therefore angle BC. And similarly it may be proved that no line but EF is GAD= angle G AF, or G A bisects the angle at BAC. Q. E. D. equal to BC; hence EF=BC. Q. E. D.

PROPOSITION XXXII.—The straight lines drawn perpendi. PROPOSITION XXXVI.-In the figure of Euclid I. 5, if ac cular to the sides of a triangle through their middle points meet (Fig. 36) be bisected in H, and co be equal to c A, then BG shall in a point.

be equal to twice BH. Bisect the sides AB, BC (Fig. 32) of the triangle ABC in From c draw cK parallel to AB, meeting BG in K (Euc. D, E, and let DG, BG perpendicular

I. 31), and join A K; then the triangles A KC, BKC are to A B, BC meet in G; draw G F per

upon the same base Kc, and between the same parallels K c, pendicular to AC; then shall F be

AB; hence they are equal (Euc. I. 37). Again, the triangles the middle point of Ac. Join AG,

AKC, G KC are upon equal bases A C, CG, and have the same BG, CG; then since AD= DB, and

vertex k; hence they are equal (Euc. I. 38); but AKC = DG is common and at right angles,

BKC; therefore triangle BKC = triangle GKC. Hence, by base AG = base G B. Similarly, be

the last proposition, BK = KG; therefore BG is double B K. cause BE= EC, and EG is common

Again, because BK = KG, triangle BKC= - half-triangle BGC; and at right angles, therefore BG=

Fig. 32.

and because Ah=HC, triangle Bhd = half-triangle BAC; GC, therefore AG=GC; and be

but because AC=CG, triangle BAC = triangle B G C, and the cause AG=GC, and GF is common, also right angle GFA= halves of equal triangles are equal; therefore triangle BHC= right angle G Fc, therefore the triangles GFA, GFC are equal triangle B K C, and the angle by Proposition XXX., and AF=FC. Q. E. D.

BC K is equal to the alter. PROPOSITION XXXIII.—The straight lines which bisect one nate angle ABC (Euc. I.29), interior and two exterior angles of a triangle meet in a point. and ABC is equal to ACB

Produce AB, AC (Fig. (Euc. I. 5). Hence the 33), sides of the triangle angle BC K is equal to the ABC, to D and F, and angle BCH. Hence the bisect the exterior angles two triangles BHC, BKC D BC, BCF by the lines have one side BC and one BG, CG meeting in g; angle Bch in the one equal join GA; then GA shall to one side BC and one

Fig. 36. bisect the angle BAC. angle BC K in the other;

Draw GD, GE, GF per- and their areas are equal, because, by Proposition XXXV., they Fig. 33.

pendicular to AB, BC, are equal in every respect; therefore, BH = BK= half BG.

CA; then in the triangles Q. E. D. GB D, G B E, because angle GBD= angle G B E by construction, Our next article will embrace the whole of Book I., and will and the right angle G DB = right angle G E B, also side GB is contain proofs of the following propositions : common, therefore the triangles are equal; therefore GD= PROPOSITION XXXVII.-If the diagonals of a four-sided GE (Eac. I. 26). Similarly GE = GF, therefore GD = GF; figure bisect each other, it is a parallelogram. and because GD = G F, and G A is common, also right angle Corollary 1.--If the diagonals be equal as well as bisecting GDA= right angle G FA, therefore, by Proposition XXX., the each other, the figure is rectangular. triangles GDA, G F A are equal, and the angle G A D equals the Corollary 2.—Hence the angle in a semicircle is a right angle. angle GAF-that is, G. A bisects the angle BAC. Q. E. D. PROPOSITION XXXVIII.-If the diagonals of a four-sided

PROPOSITION XXXIV.-If two triangles have one side, and figure bisect each other at right angles, the figure is a rhombas. one angle in the one equal to one side and one angle in the Corollary.--If the diagonals be also equal, it is a square. other, and likewise their areas equal, then shall also their other PROPOSITION XXXIX.-If a four-sided figure have its opposides and angles be equal to

site sides equal, it is a parallelogram. each.

PROPOSITION XL.--If AB, BC, CD, Da be the sides of a Let ABC, DCF (Fig. 34)

parallelogram taken in order, and points E, F, G, H be taken in be the two triangles of equal

them such that AE=CG, and BF = DH, the figure E F G H area, and let them be placed

shall be a parallelogram. so that BC, CF, their equal

PROPOSITION XLI.—If ABC be any triangle, and D E the sides, are in the same straight

line joining the middle points of the sides AB, AC, then shall line, and BCA, CFD their

De be parallel to B C, and B C shall be double D E.

Fig. 34. equal angles; then because

Corollary 1.—The angles of the triangle DFE are equal to angles BCA, CFD are equal, Ac is parallel to DF. Join AD; the angles of the triangle B AC, where F is the middle point then because ABC, DCF are equal triangles upon equal bases of BC. in the same straight line, A D is parallel to BF (Euc. I. 40); Corollary 2.—The sides of the triangle D F E are cach equal therefore A D F C is a parallelogram. Therefore, by Euclid I. 34, to one-half the corresponding sides of the triangle BAC, and A c is equal to DF; and therefore, by Euc. I. 4, the triangles are the area equal to one-fourth of the whole triangle BAC. equal in every respect. Q. E. D.

PROPOSITION XLII.--If o be any point within a triangle PROPOSITION XXXV.-If the bases of two equal triangles ABC, and D, E, the middle points of A B, AC, be joined with FG, be in the same straight line, and the line joining their vertices the middle points of o B, o C, the figure D E G F will be a paral

be parallel to this line, their lelogram.
bases will be equal.

PROPOSITION XLIII.-If in the last proposition k be the
This is evidently the con- middle points of 0 A, B C respectively, the six-sided figure
verse of Euclid I. 40. Let ABC, DKGEH shall be equal to one-half the triangle A B C.
DEF (Fig. 35) be two equal PROPOSITION XLIV.--If d be the middle point of BC, the

triangles having their bases BC, side of a triangle A BC, and if A D be joined, then if Da be с Е

EF in the same straight line, equal to D B Or D C, the angle at A is a right angle.
Fig. 35.

and let AD, the line joining

their vertices, be parallel to BF; then shall BC be equal to EF. For if BC be not equal LESSONS IN CHEMISTRY.-XXXI. to Er, one of them must be the greater. Let EF be greater, and from EF cut off Eg, equal to BC, and join DG; then

FERMENTATION. because ABC, DEG are two triangles upon equal bases, The Sugars.—The saccharine group occupies a prominent BC, eg, in the same straight line, and between the same place in Organic Chemistry. It is closely allied to the amylaparallels BG, A D, they are equal (Euc. I. 38). But by the pro- ceous group, for, as we have already seen, starch can be conposition the triangle ABC is equal to the triangle DEF; there- verted into sugar; moreover, the alcohols and their allies owe fore the triangle de F is equal to the triangle D EG, the greater their existence to the decomposition of the members of this equal to the less, which is absurd. Hence EG is not equal to group. The sugars all possess a characteristic sweet taste.

A

B

с

А

A

B

F

They are composed of carbon, hydrogen, and oxygen; the which have a hole at their apex. This hole is stopped by a plag carbon always is present in the proportion of six atoms, whilst for a few hours, and when it is removed the uncrystallisable the oxygen and hydrogen appear in the exact proportion to form syrup drains out; the loaf is finally dried, and finished in a a whole number of molecules of water. Hence the sugars have lathe for the market. been called hydrates of carbon. Their various decompositions Maple Sugar is a product of the American forests. The aro effected by the removal or addition of the elements of water; tree is tapped on its sunny side ; two holes, about half an inch when oxidised, they generally yield oxalic acid.

deep, penetrate the bark; reeds are fitted into these, and the The chief sugars are--cane sugar, or sucrose (C,H,,0); rising sap-or, if the operation be conducted in the autumn, the grape or starch sugar, or glucose, 21C,H,O,H,0); and milk descending sap—flows through the reeds into vessels placed for sugar, or lactose (

CH2.0 H,0). Besides these there are its reception. Each hole exudes as much as six gallons a day, some less prominent members of the group.

and if the tree bo old this quantity yields a pound of sugar. Cane Sugar, or Sucrose (C2H4.0u),

is the sweetening principle Beetroot Sugar.—The manufacture of sugar from the white in a great number of tropical fruits; it is chiefly procured for boet owes its rise to the wars of Napoleon. When the French the market from the sugar-cane, the maple tree, and beetroot. supply of sugar from the West Indies was cut off, they had It is also found in small quantities in carrots and turnips, and recourse to its extraction from beetroot. The expressed juice in the chestnut and pumpkin.

contains about ten per cent. of saccharine matter. The clarifiIt is soluble in water and alcohol, but ether is unable to cation, etc., is conducted apon the same principles as those effect its solution. When its water solution is evaporated, given above. it becomes deposited in crystals-oblique rhomboidal prisms Grape or Starch Sugar-Glucose (CH,,06,2,0):

This sugar --this is the sugar-candy of the confectioner. Its brown may be prepared by boiling starch in twice its weight of water, colour depends on the state to which the syrup has been acidulated with one per cent. of sulphuric acid. clarified before the crystallisation. Loaf sugar is composed Starch is C,H2,0,0, so that two molecules of glucose may of innumerable small crystals, which are transparent; its be supposed to be formed by inserting four molecules of water dazzling whiteness is due to the reflection of the light from into one molecule of starch. their many faces.

C,,H..0.0 + 48,0 = C,,H.,0,., or 2(0.5, 0.9,0). If a solution of sugar be kept at the boiling point for some time, it gradually loses the viscid nature of a syrup, exhibits an It has been shown that this change can also be effected by acid reaction, and refuses to crystallise.

diastase. It is now called inverted sugar, and its peculiar properties

This sugar has not the same power of sweetening as cane seem to be due to the fact that another molecule of water has sugar; 1 part of the latter is equal in this respect to 24 of the been assimilated, its formula being C,H,Oz If the boiling former. Neither is it so soluble in water, but is more readily still be continued for some hours more, water is taken up, and taken up by alcohol. The action of sulphuric acid readily distingrape sugar is the result, 2(CH 20,1,0). These effects are guishes these two varieties of sugar. Cane sugar is decom. produced with greater rapidity if any of the strong acids be posed by the acid, but with grapo sugar a compound is formed present in a small proportion; and hence, to prevent this -sulpho-saccharic acid. change, lime is mixed with the juice of the sugar-cane as soon milk of the mammalia. It is fitted to be an ingredient of this

Sugar of Milk, or Lactose (C,,H,,0,1,H,O), is peculiar to the as it is, as expressed.

Sugar possesses eminent antiseptic powers, and therefore it secretion by its non-fermenting quality. It may be procured is much used in " preserving." If spread over any fermentable by evaporating whey until it reaches the crystallising point or decaying matter, it will arrest the action.

Then the lactose appears as semi-transparent, right, four-sided Manufacture of Sugar.—The canes are cut close to the soil prisms, terminated by pyramids, on pieces of thread or wood, before they flower. In the rolling mill they are crushed between which are placed in the liquid to form nuclei. It has even less grooved steel rollers. The juice thus expressed is so readily sweetening power than glucose, it is more difficult of solution in fermentable that it may not be left for half an hour, but is water, and insoluble in alcohol. immediately mixed with about to its weight of lime, and heated

Fruit Sugar, or Lævulose (CH,208), which is found in ripe to 60° Cent. in copper vessels. Here the albuminous constituents fruits, seems to be a mixture of cane and inverted sugars. By a of the juice coagulate, and are skimmed off. This necessary spontaneous action the latter variety separates into crystallized heating has the effect of inverting some of the sugar, and ren

starch sugar and an uncrystallisable syrup. This may be dering it of the uncrystallising quality; and much more would noticed in the case of preserved fruits. They are covered with be so treated were it not for the lime, which neutralises a candy which does not possess a sweetness equal to that of the free acid. It is transferred from the clarifiers to shallow cane sugar. wooden coolers, and finally to the potting-casks, which have per.

FERMENTATION. forated bottoms; and in the course of four or five weeks all the A ferment is an active principle of organicised matter, which molasses--the inverted sugar-drains off, leaving "raw sugar.” is capable of setting up decomposition in organic matter. It is A gallon of juice generally yields a pound of sugar.

now generally admitted that ferments are vital organismsRefining.--If loaf-sugar be required, this raw sugar is mixed fungi, infusoriæ, etc., the very lowest forms of vegetable and with one-third of its weight of lime-water, which contains three animal life. These seedlings of life are everywhere. The very or four per cent. of "bone-black.” After being heated by in- dust which a sunbeam reveals floating in the air contains them. jections of steam, it is filtered through pipes made of cotton If their vitality be destroyed in any body, that body will not twill. Sometimes the serum of bullock's blood is added to this ferment: for instance, if milk be made to boil under a pressure syrup, which, as it coagulates, gathers in its meshes the me- of 1, atmosphere, the high temperature kills these organisms; chanical impurities.

and if the liquid be now kept from the air, or such air only bo The brown syrup is now bleached by allowing it to filter allowed to come into contact with it as has been strained by through some twelve feet of animal charcoal, which is made by passing through a plug of cotton wool, it may be kept for any calcining bones in closed vessels.

length of time without undergoing any decomposition. This clear liquid must be evaporated down to a thicker consis.

Fermentation has been classed by Miller in three divisionstency ere it will crystallise; but owing to its viscidity a tem- 1. Where the body is simply broken up into compounds of less perature of 110° Cent. is required for its ebullition. Were the complicated structure, thus :srap raised to this heat, much of it would become inverted, and much would be burnt, and so colour the sugar. The diffi

C,H,,0, (fruit sugar) gives 200, (carbonic acid) + 2C, 4,0 (alcohol). culty is removed by conducting the evaporation in a spherical 2. Where beyond this decomposition the elements of water closed vessel—the lower half of which is double-by means of are separated, thus : a powerful air-pump; a partial vacuum is produced, and when

C,H,,0, (grape sugar) = 200, +2C,H,0 +H,0. steam is injected into the cavity at the bottom of the pan, 3. Where the elements of water are assimilated, and the new the syrup boils at about 70° Cent. When it has reached a certain consistency it is placed in a vessel heated by steam to body thus formed split op into simpler componnds. Thus:170°; here it is beaten about by wooden oars until it appears

Ohne Sacar.

Fruit

Sogar. granular, when it is placed in conical vessels of earthenware,

(1) C, H,,0.1 + H,O 20 1,0,-
(2) COHO, 200, + 20.2.0.

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