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ENGLISH-LATIN.
Singular.

Plural.

Faithful friends are loved; I have great riches; they !ose wisherl-for friendship; the ground is wet; wet ground injures ;

hares have sharp teeth ; with sharp teeth we all eat; thy soldiers i Ab. || n.&v.

D.&Ab.

are brave; are thy father's soldiers brave? they delight in (abl.) credulous hope ; the horns of the bull are strong; the virtues of the king are remarkable; how beautiful is the portico ; you ought to learu Latin ; men fear the last hour; the house is guarded by a strong band; avaricious men are avoided ; ill-tempered women are never loved ; the ill-tempered are troublesome; is friendship

eternal ? hope is eternal; how slow are thy steps ! ice is slippery us, er, ir, e, er, ir,

in winter; no one loves hunger and thirst; quiet quickly flies

anay; the harbour is convenient for ships; the fearful are never um (n.)

8a fe ; art thou satisfied with the speech of thy father? they strike a poweriul prince; falling flowers are gathered (lego 3); he gainers flwers in the march; the Greek language is beautiful;

swelling seas are often found; the rest and solace of true friends 3 (various) is i (various)

-hip are wished for ; no one is always happy.

To how large an extent Latin words enter into the composi

tion of our present English is strikingly seen in the last vocaibus bulary. These words found therein have their English

representatives. u (n.)

ubus
LATIN.

E. R.
Humidus Humid, humidity

Barbarus Barbarous, barbarism, barbarity
ei

ebus

Quies

Quiet, quietness, quietly
Tumidus Zumid, tumidity, tumour

Profundus Profound, profundity
In this summary view, many facts regarding gender, num.

Nox

Nocturnal, equinox ber, and case, are of necessity omitted. They may be found

Frigidus Frigid, frigidity in the works to which reference has already been made. It

Magnificus Magnificent, magnificence seems, however, desirable to add, that grammarians recognise

Commodus Commodity, commode in Latin what is called a common gender. Those nouns are

Lubricus

Lubricate said to be of the common gender (c.), which may be applied

Felix

Felicity, felicitate indifferently, either to a male or a female. Such nouns are

Credulus Credulous, credulity, incredultiy hospes, a guest ; hostis, an enemy; incola, an inhabitant;

Clarus

Clear, clearness, clarify parens, a parent ; sacerdos, a priest or priestess ; testis, a

Gelidus

Jelly witness ; bos, a bull or a cow; canis, a dog or a bitch; lepus, a

Gradus

Grade, graduate

Potens hare ; mus, a mouse, &c.

Potent, potentate, potency
Avárus

Avaricious, avariciously
GENERAL EXERCISES ON THE Five DECLENSIONS.

Fames

Famish, famine

Rotundus Rotund, rotundity
VOCABULARY.

Infidus

Infidel, infidelity

Sempiternus Serpiternal Húmidus, a, um, humid, wet; hiems, hiemis, f. winter; divitiae,

Tardus

Tardy arum, f. riches ; ultimus, a, um, the last; lepus, dris, m. a hare;

Contentus Content, contentedness cadacus, a, um, falling. frail; pavidus, a, um, fearful, timid ; barbarus,

Limpidus Limpid a, un, barbarous ; sermo, ónis, m. speech; Látinus, a, um, Latin;

Acutus

Acute, acutely Graecus, a, um, Greek; exoptatus, a, um, wished for, desired'; quies,

Morosus

Morose, moroseness quietis, 1. rest, quiet: tumidus, a, um, tumid, swelling; profundus, a,

The student of Latin will be greatly assisted, if before he um, deep; inspiratus, a, um, unhoped for; nox, noctis, f. night; frigidus, a, um, cold; magnificus, a, um, magnificent ; ligneus, a, attempts to commit a Latin word to memory, he tries to find um,, wooden; commodus, a, um, convenient ; glacies, ei, f. ice; an English word which is derived from it, and with which he luhtivus, a, um, slippery; nemo, neminis, c. no one; felix, felícis, may associate it in his mind. happy; credulus, a, um, credulous, too believing; palus, paludis, f. a marsh; clarus, a, um, clear, distinguished; gelidus, a, um, cold; gradus, ùs, m. á step; potens, potentis, powerful; nunquam, adv. never ; avárus, a, um, avaricious; fames, is, f. hunger; bilis, is, f.

LESSONS ON PHYSIOLOGY.-No, V. thirst; rotundus, a, um, round : infidus, a, um, unfaithful; sempiternus, a, um, everlasting; tardus, a, um, slow; contentus, a um,

MAN. satisfied; limpidus, a, um, limpid, Bright;, exiguus, a, um, short, We wish to make our lessons so simple and plain, that you narrow; acutus, a um, sharp; humus, i, f. the ground or soil; may thoroughly understand them. In treating, of digestion, eximius, a, um, eminent, remarkable ; morosus, a, um, morose, ill

we showed you that the process begins with taking our food lempered; semper, adv. always.

into the mouth. It is there masticated or chewed ; in being LATIN-ENGLISH.

chewed, it becomes mixed with the saliva of the mouth, until Est mihi amicus fidus et carus ; infidus est servus tuus ; terra est it takes on the character of a soft pulp. It is then swallowed; rotunda; vera amicitia est sempiterna; fames et sitis' sunt mos passes down the æsophagus into the stomach, where it comes lestae ; avárus nunquam est contentus; rex est potens ; gradus in contact with the gastric juice. By the action of this fuid, it tuus tardus est; virius patris tui est eximia; fons est clarus et is converted into chyme. This chyme passes gradually into gelidus ; nomen clarum est ducibus ; amnis limpidus delectat the duodenum, where it meets the bile and the pancreatic omnes; cervo sunt alta cornua ; res est magna et insolita; hic sunt juice, undergoes a complete chemical change, and reappears in vastae paludes; opes credšla fallit pueros, hominibus exigua est the form of a white milky liquid, known by the name of whyle. dies; nemo semper felix est; glacies est lubrica ; pons ligneus This is taken up by the lacteals, and by them poured into the custoditur ; non omnes milites sunt fortes ; magnificae porticus thoracic duct or canal, which is in communication with the defenduntur; portus est commodus; dentibus acutis edimus; nox greater portion of the absorbent vessels of the body. In this est longa et frigida; bonus laudatur, improbus vituperatur ; canal the chyle becomes mixed with the lymph, and is thence senectus saepe est morosa ; insperata salus venit; mare vastum, profundum, tumidum; quies valde exoptata facile amitti: conveyed into the vein which passes under the left clavicle, tur; sermonem Latinum discimus; nonne doces Graecam linguami and mingling with the venous blood, is carried to the heart gentes barbarae remotae sunt; lepores pavidi evolant ; los est and lungs to receive its vital and life-giving properties. caducus ; hora ultima venit; incertae sunt divitiae ; mores antiquos

If the food which we eat is thus at last converted into blood, amat mater mea; verba tua sunt dura ; quam humida est humus! and if it be the blood which builds up every individual part of Qon facile in hieme agri arantur.

the body, it is of great importance that our food should be good and wholesome. The blood can contribute to growth and common egg-shell, which is nothing more than so many layers nealth only as it exists in a pure and healthy condition. If of fibrous tissue enveloping the albumen, and forming is thus precious vital fluid be itself diseased, it cannot fail to com- thin membrane which comes between the outer shell and the municate the disease to the various parts or organs which it inner substance. These fibres constitute the first and simples has to supply with their appropriate iautriment. For example: forms of animal tissue. If this solid earth, which we wait -If a fit of passion may suddenly and immediately occasion with so firm a step, be but an aggregation of particles or atoms, such a change in the milk of a nurse as to render it a rank held together by the one great law of attraction, our bodies are poison to the little dependant infant, there is nothing to con- nothing more than a mere combination and union of elements tradict the theory, that the blood itself may undergo such under the law of organisation. Nor is it difficult to become changes as to convert it from a wholesome nutriment and in some degree acquainted with these elementary or compostimulus to vital action, into a most violent puison, fatal nent parts, with their physical, chemical, and vital properda. even to life itself. Another example:-In vaccination, a Since the growth of the cell is dependant on its absorbing surgeon introduces into the arm of a child a very minute por certain particles of matter from the duid which surrounds tion of virus, which in some way or other, not well known, --in this nutrient fluid, in the process of organisation, or before affects and alters the whole of the blood; and this morbid the process begins, we must look for the components of the state of the blood continues for a length of time. Or, again, animal structure, with their essential or peculiar properties suppose a student in the course of dissection should prick his But the blood is not more dependant on the character of the finger, the putril matter thus introduced may so effectually food which we eat, than on the purity of the air which we get into his system as to poison the blood and occasion death breathe. The heart, from which the blood issues in a coaditself. It fotows

that a man may wilfully and knowingly tion to nourish the body, is situated between the right and se induce disease, and injure his system. A drunkard is never a left lung, and with the lungs fills up the whole cavity of the healthy man. Some men may more easily and for a longer period chest, as may be seen by the accompanying cut. Bach lang resist the effects of intemperance than others; but that the free is made up of a countless number of salla or vesicles, which use of ardent spirits is prejudicial to health is a truth which all the facts of physiology but too clearly demonstrate. An intemperate man does everything to contravene nature. He is working against God, and against the most beneficent laws of his universe. The great Creator has introduced into the blood all those elements which are adapted to preserve it pure and uncorrupt. What other end can we conceive to be in. volved in the fact, that in the blood is to be found a certain portion of saline matter? The presence of such an agent in the circulating fluid must be regarded as a beautiful and beneficent provision to prevent its decomposition. Were the blood to decompose in the body, it would cease to possess any vital property; and, deprived of its vitality, it could no longer minister to the nutriment and growth of a single structure.

The blood is a liquid of a beautiful red colour, and of a peculiar odour. In some animals this odour is very marked. Take blood from a cow, and by the smell of the fluid you can tell from what animal it has been drawn. In its living state, the blood is a transparent liquid, folding in suspension certain little bodies, of which some are colourless, but the greater portion of which have a red colour, and are known by the name of blood-globules. Now, to under. stand how these little bodies are adapted to nourish and build up the body of the strongest and most powerful, we shall try to set before you the compo. Bloodnent parts of this precious fuid. Let us open a globule. vein, and take from the body a portion of blood. If we allow it to remain at rest for ten or fifteen minutes, it begins to congeal and take on a more solid form like that of a soft jelly. The fluid has become a solid, and this is the only change which is yet palpable to our senses. After a few hours we ånd that are always full of air derived by inspirations from the atmo the clot has a greater degree of consistence, and, as the effect sphere which surrounds us. These cells communicate freels of this contraction, is surrounded with a transparent yellow the one with the other; and it is in these air-cells of the luap Auid, which is named serum. Now what is there in this blood that the dark venous blood, by coming into contact with the to produce this coagulation : Why does it not remain in a oxygen, is converted into arterial. Impure air, therefore, esse Auid state, as when first drawn from the vein? There must be not but be prejudicial to the quality of the blood, and in the some peculiar law to account for this change. In itself, and degree in which the blood is affected, must the body, with as it is seen flowing in the veins of a living creature, it appears its peculiar functions, be more or less disordered. No one a colourless fluid, with minute red particles which give the should sleep in a room into which air has no admission, fors blood its beautiful scarlet hue; and so long as it is in a fuid is possible that during the night he may exhaust the whole of state it holds in solution a particular substance called fibrine, the vital air necessary for respiration contained in the apart which, in its ultimate composition, differs little or nothing ment, and the consequence must be suffocation. Nor should from albumen, or the white of an egg. This substance is dis- we leave our sleeping-room in the morning without throwing tributed through the whole body, but is found chiefly in the open the window and allowing a free current to pass through blood, because the blood, in its course and flow, supplies to every it. Not only is it important to eat food which is wholesceus individual part of the complex structure the materials of its and easy of digestion, but to breathe the freest and the paret growth and development. Take the blood from the living struc- air. No one should choose a house in a crowded, con ined ture, and the fibrine remains no longer in solution. Instead of and thickly-peopled neighbourhood. The atmosphere of such being diffused, it coagulates and contracts, till it has pressed out a neighbourhood is always more or less impure. Every inspi the serum by the mutual attraction of its own particles. Now if ration which we take, or every breath which we draw, we take we look at this clot or congealed blood through a microscope, a portion of this air into the lungs, and the blood in its pas we shall find that it presents a peculiar arrangement. It is sage through the lungs so filled, must to a certain extant de not a mere aggregation or promiscuous accumulation of parti- come tainted, and may set up disease in the system. Hence cles, but a beautiful disposition of fibres crossing one another the importance of daily exercise in the open air. The farther in every direction. This arrangement may be seen in the we can get away from the smoke, and dust, and surcharged stree

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sphere of large and crowded towns the better. God made the What is the acid which they throw off, and what is the element
coustry, and no one can ramble for a single day amid its fields which they take from the air
and woods, its hills and vales, without feeling the stream of What is the effect of impure air on the health?
life bounding through his system with a freer, fresher,

Show wherein it is injurious.
mightier, and more living force than before.

Give an example of the bad effect of an excess of carbonic acid

on inferior animals.
All organised bodies are subject to decay, even while actively
engaged in performing the functions of life. One of the chief
products of that decay is carbonic acid, the removal of which
from the system is essential to health and vital action. This

LESSONS IN GEOMETRY.No. IV.
removal is affected by the agency of the lungs, which are made

.PROBLEMS IN PRACTICAL GEOMETRY-Continued. up of a countless number of little air. cells. Now between the atmospheric air,

10. Given two angles of a triangle to find the third angle. Draw which is taken into the lungs at every

a straight line D F (fig. 11), and take any point e in it. Then,
breath which we draw, and the blood, at the point e, in the straight line e d,
there is nothing interposed but a very

make the angle dec equal to one of the
thin and delicate membrane. The carbon given angles; and at the point e in the
discharged from the lungs of a man in straight line Ec, make the angle ceu
good health is from eight to nine ounces equal to the other given angle; and h E F,
in the twenty-four hours. Let a hundred will be the third angle required.
such individuals be shut up together in a

The reason of this construction is plain,
room for three hours, and during that space of time not fewer from Prop. XXXII., Book I., Euclid, which
than one hundred ounces of carbonic acid must be evolved ; and asserts that the three angles of every triangle are equal to two right
what is the consequence? The air which is breathed being im- angles; and from Prop. XIII., Book I., which

asserts that the pregnated with carbonic acid, the quantity of carbon exhaled, angles which one straight line makes with another upon one side becomes, with every successive respiration, less and less. It has of it, are either two right angles, or are together equal to two been found by experiment, that, when fresh air is taken in at right angles. each inspiration, thirty-two cubic inches of carbonic acid may

The proof of the latter proposition is very easy; for by be exhaled in a minute; while, in those cases in which the referring to figs. 12 and 13, it will be seen at once, that the same air is breathed repeatedly, the quantity of carbonic acid

Fig. 12.

Fig. 13.
emitted in the same period may not amount to one-third.
And if it be true that however often the same air may be re-
spired-should it be respired till it is no longer capable of
sustaining life-it never becomes charged with more than ten
per cent.

of carbonic acid, we see the necessity of a constant
supply of fresh air by means of free ventilation. The discom-
fort inseparable from a crowded audience in a church, lecture-
room, or theatre, which is not provided with sufficient venti angles which one straight line makes with another are
lation, is due in great part to the continued respiration of the either right angles or oblique angles ; and that the oblique angles
same air, which has become loaded with carbonic acid. - Nor
can it be denied that the habitual respiration of such air in the are equal to the right angles is plain, because they occupy
narrow and pent-up dwellings of the poor, or in crowded it is evident in fig. 13, that whatever the angle da В wants

exactly the same space about the point A; in other words, factories and workshops, has a tendency to impair health, of a right angle, Se angle CAB has just exactly that quantity diminish mental activity, and even deteriorate the moral sensibilities.

more than a rigt: angle; and consequently the two to. Repeated experiments have brought out the fact, that the jether as just equal to two right angles. From this argu. blood comes to the lungs charged with carbonic acid, which it ment, it is likewise evident, that whatever be the number o gives up, and receives oxygen in its stead. According to the line or (fig. 11 above), on the

same side of it, all the angles

straight lines which meet at the same point E, in the straight law of mutual diffusion, the quantity of oxygen absorbed, or which they make, taken together, are only equal to two right drawn from the atmospheric air, exactly replaces the quantity of carbonic acid get free. But for this change in the compo

angles. sition of the blood, all the functions of life would soon be so easy; but a palpable proof of it may be obtained by any

The proof of the 32nd proposition of Book I. is not quite arrested. Put a fish in water which is impregnated with car- student for himself as follows :-First, accurately construct & bonic acid, and its death is almost instantaneous. Introduce the buman being into some crowded room, in the atmosphere triangle of any

kind, on a piece of paper or card, and then cut of which there is wanting a due proportion of oxygen, or in three angles of the triangles at about half the length of each

it out entire ; next draw a straight line, and cutting off the fainting ensues. Let the blood be denied its contact with the side, apply their three angular points to any point in this atsapheric air, and life will speedily ebb and disappear. straight line, so that the legs

of the angles shall be contiguous in The blood stagnates in the small capillary vessels of the lungs guous to the straight line;

when it will be found that the

contact with) to each other, and one of the exterior legs conti-the heart has not sufficient force to drive the blood through other exterior leg is also contiguous to it; thus showing that those capillaries--the air included in the lungs loses more of the three angles ill up the space of two right angles, and are its oxygen, and becomes overcharged with carbon, and at last therefore equal to two right angles. the blood becomes so venous, that the pulmonary circulation is altogether suspended, and death follows.

Another mode of proof may be had from the use of the pro

tractor, fig. 14, described at p. 50, No. 4. Draw any triangle QUESTIONS FOR EXAMINATION.

Fig. 14.
('an you describe the process of digestion ?

N
How is health affected by the condition of the blood ?
Give an example of how the blood becomes tainted or corrupted ?
What bodies does the blood hold in suspension ?

What change takes place in the blood after it is dra from the
veio ?

What is the peculiar substance in the blood which goes to build up the various tissues of the body?

Why should we be careful as to the character of the food which we take?

How is the blood affected by the air, and how does it come into contact with the atmosphere

of what are the langs composed, and where are they situated ?

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as before ; measure each of the three angles by the prouaetor Ave sides,- viz., A B, BC, C D, D3, and 5 s; and turn fave sides in degrees ; add the nuo.bers of the degrees obtained by these, are ten sides. Now, taking away four right a: zlts from len three m-asuremeres together; and if they have been carefully right angles, leaves siz right angles; and this is the number of incasured, they will always make 180° ; !or one right angle right angles to which the tire angles of the figure, -5.2., 4 BC, makes 90°, t..ercfure two right angles make 180°. Thus, if B C D C D E, D E A, and s A B, are, to getr.er, equul. Tho one anule of a triangle measures 57°, and another measures proof of this is very evident, for, by draw.

Fig. 18. 75°, then the third must measure 48°, because 57+75+45=180. ing straight lines c'À, CE, from any angle A

From the 13th proposition of Book I., the 15th is easily in- to the other angles of the higure, there are ferred; viz.., that if two straight lines intersect each other, the three triangles, ABS, A CE, and C E D vertical or opposite angles are equal. One angle is said to be formed within it. Now, by the 32nd protertical or opposite to another, when the legs of the former position, all the angles of these triangles proceed from the point of intersection or meeting, in directions are equal to twice three right angles, that exactly opposite to those of the latter. Thus in fig. 15 let the is, to six right angles, as before. The same straight lines BE, CD, intersect each other in the point a; then thing could be proved of any other polygon, the angle BAD forned by the production of the legs EA, CA, B C D F G. In like manner, whatever be Fig. 15.

of the angle c A r, in opposite directions A B, the number of sides in the polyg n, it may be shown that the A D), through the point of intersection a, is truth of the proposition is manifest, for the addition of another said to be rertical or opposite to the side to the figure would only add another pair of right angles to angle CAE. In like manner the angle CAB the number of right angles it presiuusly contained ; 80 that,

is said to be opposite or vertical to the whatever was the number of sides, there would always be four c angle D A E. The proof of this proposi. right angles wanting in order to make up twice as many right

tion is manifest, for by turning over the angles as the figure has sides. This truth may be exhibiied angle Bad upon the angle CA E, they would in another way. By drawing straight lines from any angle in coincide, and they are therefore equal. a polygon to the other angles of the figure, we draw as many

Otherwise it is shown that they are equal, straight lines as the figure has sides, waring trco, seeing that, because either of them together with the angle Bac makes in drawing these straight lines, we draw two that must coincide two right angles.

with two of its sides already drawn; hence we construct in the Depinition 1.-When two angles are together equal to two interior of the figure as many triangles as the tigure has sides, right angles, the one angle is said to be the supplement of the wanting two. It is plain, therefore, that it will then contain other. Tous, in the preceding figure, the angle B A C is the only twi e as many right angles as the figure has sides, wanting upplement of either of the angles BAD OR CA P. In like your right angles. manner, the angle B A D is the supplement of either of the From this corollary another may be drawn regarding regular angles BAC O DAE. If the angle BAD contained 30°, polygons. Since all the sides and angles of regular polygons are then the angle BAC must contain 150°.

equal to each other; that is, the sides equal to one another, and DEFINITION 2.—When two angles are together equal to one the angles equal to one another, it is plain that the value of tach right angles, the one angle is said to be the complement of of the angles can be determined. Thus : double the number of the other. Thus in fig. 16, the three

Fig. 16.

sides, call them right angles, and subtract four right angles; angles of the triangle E B D are

the remainder will be the number of right angles it contains ; equal to two right angles; but the

divide this remainder by the number of sides, and the quotient angle E B D is a right angle by con

will be the value of each angle. Example: In a regular pen. struction ; it, therefore, follows that

tagon there are 5 sides; twice five are 10; from 10 right the other two angles B E D, B D E

angles, subtract 4 right angles, and 6 right angles remain; are together equal to a right angle,

B divide 6 by 5, and the quotient is one and one-fifth; hence, and the angle B E is the comple

every angle in a regular pentagon is equal to one right angle ment of the angle B D E; or the

and one-fifth of a right angle, or 108°. angle B Dr is the complement of

The second corollary which Euclid draws from the 32nd prothe a, gle B ED. If the angle B de contained 40°, then the position, is that all the exterior angles of any rectilineal figure or angle B E D must contain 50%.

polygon are together equal to four right angles. By the exterior From the propositions above mentioned, it evidently follows, angies here, is meant those which arise from producing each of that when two straight lines intersect each other, the four angles the sides of the figure in one direction only, say from right to thus formed at the point of intersection, are together equal to four left, in going round the figure. The truth of this proposition right angles; that is, in rig. 15, the four angles B A c, b A p, is manifest, from the consideration that through any point ex. D A E, and A C, are together equal to four right angles. It ternal to the polygon straight lines can be drawn parallel to is also evident, that all the angles formed by any number of the productions of the sides, and forming with each other straight lines meeting together in a point, are together equal to angles equal to those formed by the production of each side, four right angles. Thus, in fig. 17, let any

Fig. 17.

and the adjacent side; now all the angles which can be formed number of straight lines AO, BO, CO, D 0,

by any number of straight lines meeting in a point are equal 10, meet together in the point o; then all

to four right angles ; therefore, all the angles formed by the the angles, A O B, BOC, COD, D O E, and

production of each side, and the adjacent side,--that is, all the IOA, are together equal to four right

exterior angles of any polygon, are together equal to four right angles.

angles. This might also be inferred from another considera. From the 32nd proposition, Book I.,

tion,-viz., that as the magnitudes of the exterior angles would Euclid, two very important corollaries are

not be altered by the size of the figure, provided its interior deduced, which we now proceed to ex

angles remained of the same value, neither would they be plain, as they are necessary to be understood in connexion altered if the figure were ever so small, say reduced to a with some practical problems. But first we must explain point; and the exterior angles would then be the angles about what is meant by corollary. The Latin word corollarium, from a point, and therefore equal to four right angles. which it is derived, signifies an overplus, or surplus, or something in addition which may be considered as a gratuity. In

QUESTIONS ON THE PRECEDING Lesson. geometry it means an inference or deduction easily drawn from a

How many right angles are equal to all the angles of any triangle proposition just demonstrated. The first corollary which taken together? How is it proved mechanically that all the angles Euclid draws from the 32nd proposition is, that all the interior of any triangle are together equal to two right angles? If one

straight line makes angles with another on one side of it, how angles of any convex rectilinear figure or polygon are, together, equal many right angles are they equal to? If on both sides of it how to tu ice as many right angles as me figure or polygon has sides, many? What is the supplement of an angle? What the com. wanting four right angles. Thus, in the polygon A B C D E, plement? What is the meaning of the word corollary? What fig. 18, all the interior angles ABC, B C D, CDE, DE A, and are the two very important corollaries which Euclid has drawa LA B, are, together, equal to twice as many right angles as the from the 32nd Prop of bis 1st Book ? How are they demonstrated ? ngure has sides, wanting four right angles; for this figure has What is the value of each angle of a regular hexagon?

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GEOMETRICAL THEOREMS.

curse, became, by the ever-merciful goodness of God, “ the The demonstration of all the six cases of the Pythagorean mother and nurse of the human race. It is the source of the theorem has been received from several of our mathematical most solid wealth, and of the most exhaustless treasure. For correspondents. As the extension of the 14th proposition of nearly six thousand years has this little earth of ours been the First Book of Euclid, adverted to in No. 7, has not been tilled, and sown, and reaped, and yet never was its produce understood, owing to the mode in which it is expressed, we richer or more abundant than at this moment. “Though mines here give the enunciation in a more distinct form, and add the of gold and silver should be exhausted, and the moneys made demonstration of it. We consider it as a lemma; that is, a of them lost; though pearls and diamonds should remain hid preliminary proposition to the extension of the 47th proposi in the womb of the earth and sea; though commerce should tion of the Firsi Book.

be prohibited; though all arts, which have no other object If at a point in a straight line two other straight lines make the than embellishment and splendour, should be abolished; the two angles on opposite sides of it, together equal to two right fertility of the earth alone would afford an abundant bupply" angles, these two straight lines shaú be in the same straight line; for every, possible condition of the world and of society. but if they make two right angles on the same side of it, they shall Hence it is that agriculture was held in such esteem among coincide with each other.

the ancients. It was the earliest employment of man; and At the point a, in the straight line c p, fig. 1, let the straight the wonderful length of life which those enjoyed who lived line A B make the two angles B A C, B A D, on opposite sides of it, before the flood, could not be other than favourable to the equal to two right angles, then AC, Á D, are in the same progress and improvement of this interesting art. Nor is there straight line.

anything very extravagant in the conjecture, that, as Noah Fig. 1.

For if a d is not in the same straight laid up food in the Ark for himself and those who were with line with a c, let A E be in the same him, he was also careful to preserve various seeds of corn and straight line with it. Then, by the 13th vegetable with which to sow the earth when the waters should

proposition of Book I., the two angles, bac, subside; whilst his knowledge of reaping, sifting, and grinding B B A E, are equal to two right angles'; but the grain, would be made the common property of his family the angles B A C, B A D, are, by hypothesis

and their posterity. (i.e., supposition), equal totwo rightangles;

The Egyptians, the most ancient people on earth who have therefore the two angles B A C, B A E, are,

a history and an historical life, seem to have been well acby Axiom I., Book I., equal to the two angles B A c, B’AD; quainted with agriculture. Not only was the population very take away from these equals the common angle B c, and numerous, but the land was so well cultivated as to be able to by Axiom III., Book I., the remaining angle BA e is equal support its own inhabitants, and to export large quantito the remaining angle B A D; that is, the less equal to the ties of grain to other countries. Its wonderful fertility greater, which is impossible; therefore A E is not in the same

was owing chiefly to the overflowing of the Nile, that largest straight line with a c. In the same manner it may be shown and noblest river in the old world, and the only one that flowed that no other straight line can be in the same straight line from ten to twelve hundred miles without receiving a single with a c but A D; therefore a d is in the same straight line tributary stream. A distinguished traveller has thus described with A O.

the phenomena connected with the overflow of this great Again, at the point a in the straight line cd, let the straight river :-" The air is so much rarefied by the sun during the line 4 B make the two angles B A E, BA D, on the same side of it, time that he remains almost stationary over the tropic of two right angles, then ad, a e, coincide with each other. For, Capricorn, that the winds, loaded with vapours, rush in upon if possible, let them not coincide with each other. Produce a D

the land from the Atlantic Ocean on the west, the Indian toC; then, by the 13th proposition of Book I., the two angles Ocean on the east, and the cold Southern Ocean beyond the BAD,

BAC, are equal to two right angles ; but the angle ba vis, Cape. Thus a great quantity of vapour is gathered, as it by hypothesis, a right angle; therefore, the angle bacis a right were, into a focus; and, as the same causes continue to angle; now, by hypothesis, the angle B A B is also a right angle; operate during the progress of the sun northward, a vast train therefore, the two angles Ba B, Ba C, are equal to two right of clouds proceeds from south to north, which are sometimes angles; wherefore, by what was demonstrated above, A C, AE, extended much farther than at other times. In April, all the are in the same straight line; but, by construction, A C, A D, are rivers in the south of Abyssinia begin to swell; in the beginalso in the same straight line ; therefore, the straight line Ai ning of June they are all full, and

continue so while the sun coincides with the straight line a D; for it A B, A D, do not coin- remains stationary in the tropic of Cancer. This excessive cide, they have a common segment A c, which, by Euclid's rain, which would

sweep off the whole soil of Egypt into the corollary to the 13th proposition of Book I., is impossible;

bea were it to continue without intermission, begins to abate wherefore the straight lines As, AD, coincide with each as the sun turns southward; and, on his arrival at the zenith other.

of each place, on his passage towards that quarter, they cease Remarks: --That the latter case of the preceding proposition entirely: Immediately after the sun has passed the line, is absolutely necessary for the clear understanding of the various he begins the rainy season to the southward. The rise of cases of the Pythagorean theorem, we hold to be undeniable. the Nile at Cairo, does not commence till June, the green Otherwise, how are we to know that the sides of the two colour produced either by the influx of corrupt or stagnant smaller squares a C, A G, fig. 2, page 110, No. 7, coincide with water, or by the action of the hot south winds on the sluggish each other. It might be stated indeed as a sufficient reason stream, appearing about the twelfth day of that month. "The for this, that all right angles are equal to another ; for if the red appearance, occasioned by the arrival of the Abyssinian one angle did not coincide with the other, then, of course, waters, takes place early in July, from which the rise of the one of them must be greater than the other, which is contrary river may property be dated, as it then begins to increase to the hypothesis. This to many would appear a sufficient rapidly. By the middle of August, it reaches half its greatest demonstration.

height, and it attains its maximum towards the end of Septem

ber. From the twenty-fourth of that month, the waters are LESSONS IN ANCIENT HISTORY.–No. V.

supposed to decline, but maintain nearly the same level till

the middle of October. By the tenth day of November, they By ROBERT FERGUSON, LL.D.

have sunk about half, and from that period continue to subBy the conquest of Cambyses, the far-famed land of Egypt side very slowly till they reach their minimum in April.” became a province of the Persian empire. But, before we Some writers have fallen into the error that this periodical pursue the history of conquest any farther, it will be both in- ) inundation is the only means which the greater part of the teresting and instructive to turn our thoughts to the state of country po-sesses for irrigation or being watered; but when the arts and sciences at this earliest period of human and it is said that no rain falls in Egypt, it must be understood of earthly development.

Upper Egypt, or the Thebaid ;-in Lower Egypt, and on the Among those arts the first place must be given to AGRICUL- sea-coast, the rains are frequent and heavy. The land is not TURE. This is the most ancient as well as the most useful. destitute of water, and the dewe fall rich and copiously. Dr. After his fall, man was sent forth TO TILL THE GROUND; and Clarke says—"The vegetation of Egypt-even the redundant thus what in itself bore the nature of a punishment and a produce of the Delta—is not owing solely to parcial inundation

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