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Sagittate Leaf (Fig. 27).-A leaf shaped like the head of an arrow, from the Latin sagitta, an arrow, triangular in form, with pointed lobes at the base extending backwards. A variety of this form is called hastate, or spear-shaped, from the Latin hasta, a spear.

Spatulate Leaf (Fig. 28).-A leaf formed something like a spatula (Latin, spatula), a broad flat knife used by chemists for spreading plasters. It is broad and rounded at the end, but tapers gradually towards the stalk.

Verticillate Leaves (Fig. 29).-When more than two leaves

LEAF. 39. PINNATIFID LEAF. 40. PALMATE LEAF. 41. 45. CILIATE LEAF. 46. SERRATE LEAF. 47. OVAL LEAF.

51. ACUTE LEAVES.

grow on the same level, they are termed verticillate, from the Latin verticillus, the whirl of a spindle, derived from verto, to turn. Leaves growing in this manner, in a ring round the stem, are also said to be whorled.

Pinnate Leaf, with Tendrils (Fig. 30).-Here we have two opposite leaflets, with a tendril issuing from the point of junction between them. Found in the leaf of the everlasting pea.

Cordate Leaf (Fig. 31).-A leaf, such as the leaf of the limetree, so called from being shaped like a heart, from the Latin

cor, cordis, the heart. A cordate leaf is broad at the base, where it is attached to the petiole, and pointed at the extremity. When a leaf is narrow or pointed at the base and broad at the end, or shaped something like the figure presented by the section of a pear, it is called obcordate.

Confluent Leaves (Fig. 32).—Leaves which are joined together, or which surround the stem in such a way that it appears to pass through the centre of them; from the Latin con, together, and fluo, to flow. Leaves of this kind are more often called perfoliate.

Lanceolate Leaf (Fig. 33).-A leaf formed like the head of a lance, oblong, narrow, and tapering from the broadest part in the centre towards the base and extremity.

Orbicular Leaf (Fig. 34).— A leaf circular in outline, from the Latin orbiculus, the diminutive of orbis, a globe or sphere. Leaves of this kind resemble peltate leaves in shape, but differ from them in being cleft as far as the point of junction with the petiole. A good example may be found in the leaf of the common mallow.

Dentate Leaf (Fig. 35).-When the edge of a leaf is notched or indented it is said to be dentate, from the Latin dens, a tooth. When the margin of the leaf is unbroken, as is the leaf of the myrtle, or nasturtium, it is said to be entire.

Deltoid Leaf (Fig. 36).-A leaf with a broad base and triangular in form, so called from its resemblance to the Greek letter A, or capital D, called delta.

Decomposite Leaf (Fig. 37).-A leaf divided into a great number of leaflets, as in the illustration, in which leaflets are attached on either side to the branches which issue from the petiole. It should be noted that the meaning of this term is very different to decomposition, which means a state of decay or dissolution, the word decomposite being derived from the Latin compono, to put together, with de prefixed to increase the force of its signification, and indicating a composition of things already compounded, the leaflets of the compound leaf being also themselves compound.

Reniform Leaf (Fig. 38).-A leaf shaped like a kidney, and so called from the Latin ren, a kidney.

Pinnatifid Leaf (Fig. 39).-A leaf indented along the margin with deep irregular notches extending about half way into the mid-rib, as in the leaf of the dandelion, or sowthistle; so called from the Latin penna, a feather, and findo, to split.

Palmate Leaf (Fig. 40).-A leaf consisting of five leaflets attached to a common petiole, so called from its resemblance to the extended fingers of the hand, from the Latin palma, a hand. Leaves of this kind are sometimes termed quinate.

Digitate Leaf (Fig. 41).-A leaf consisting of several leaflets or lobes proceeding from the same point of a common leafstalk, so called from the Latin digitus, a finger, the lobes being extended like the fingers of a hand. An example may be found in the leaf of the horse-chestnut. Scarcely differs from the last. Capillary Leaf (Fig. 42).-A leaf branching out in all directions in narrow hair-like divisions, so called from the Latin capillus, hair. Examples of this kind of leaf are found in some of the tribe of umbelliferæ.

Spiny Leaf (Fig. 43).—A leaf with spines or sharp points projecting at intervals round the margin, like the leaf of the holly, so called from the Latin spina, a thorn.

Sessile Leaves (Fig. 44).—When leaves are attached to the stem of a plant without any petiole or leaf-stalk, they are termed sessile, from sessum, a part of the Latin verb sedeo, to sit, because the leaves are closely attached to the stem as if sitting on it.

Ciliate Leaf (Fig. 45).-When a leaf is bordered or edged with short hair-like appendages it is termed ciliate, from the Latin cilia, eyelashes.

Serrate Leaf (Fig. 46).-When the margin of a leaf is toothed sharply, like a saw, the teeth projecting forward, as in the roseleaf, it is termed serrate, from the Latin serra, a saw.

Oval Leaf (Fig. 47).-A leaf longer than it is broad, but equally rounded at the base and extremity, so called from the Latin ovum, an egg. Oval leaves which are broader at the base, where the leaf is attached to the petiole, than at the extremity are called ovate; but leaves which are narrower at the base than at the extremity are called obovate.

Pinnate Leaf (another variety). (Fig. 48).-Consisting of pairs of leaflets ranged along a common petiole opposite to each other, and attached to the common petiole by leaf-stalks; so called from the Latin penna, a wing, the attachment of each pair being like the wings of a bird, or the small feathers that branch out on either side of the mid-rib of a complete feather.

Bipinnate Leaf (Fig. 49).-A leaf consisting of pairs of pinnate leaves arranged along a common petiole opposite to each other; the leaf, in other words, being pinnately branched, and each branch pinnate with leaflets Leaves are tri-pinnate, or three times pinnate, when the mib-rib is pinnately branched, the branches again pinnately branched, and these last furnished with leaflets pinnately arranged.

Distichous Leaves (Fig. 50).—Leaves springing from alternate points in two rows, one on the right of the stem, and the other on the left, from the Greek dioTixos (pronounced dis'tick-os) a couplet.

Acute Leaves (Fig. 51).—Narrow leaves terminating in a sharp point, from the Latin acutus, sharp.

The above list includes the principal terms applied to leaves. Sometimes, however, to describe a leaf correctly, it is necessary to apply two or three of these terms; as, for example, when a leaf is long, narrow, and pointed at either end, fringed with hair-like appendages, and notched with small regular indentations along the margin projecting forwards, it is described as lanceolate ciliate serrate.

READING AND ELOCUTION.-V.
PUNCTUATION (continued).

VII. THE PARENTHESIS, CROTCHETS, AND BRACKETS.
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41. A PARENTHESIS is a sentence, or part of a sentence, enclosed between two curved lines, thus ( ).

42. The curved lines in which the parenthesis is enclosed are called Crotchets.

43. The parenthesis, with the crotchets which enclose it, is generally inserted between the words of another sentence, and may be omitted without injuring the sense.

44. The parenthesis should generally be read in a quicker and lower tone of voice than the other parts of the sentence in which it stands.

45. Sometimes a sentence is enclosed in marks like these [], which are called Brackets.

46. Sentences which are included within crotchets or brackets, should generally be read in a quicker and lower tone of voice. 47. Although the crotchet and the bracket are sometimes indiscriminately used, the following difference in their use may be noticed:-Crotchets are used to enclose a sentence, or part of a sentence, which is inserted between the parts of another sentence; brackets are generally used to separate two subjects, or to enclose an explanation, note, or observation, standing by itself. When a parenthesis occurs within another parenthesis, brackets enclose the former, and crotchets enclose the latter.

Examples.

I asked my eldest son (a boy who never was guilty of a falsehood) to give me a correct account of the matter.

The master told me that the lesson (which was a very difficult one) was recited correctly by every pupil in the class.

When they were both turned of forty (an age in which, according to Mr. Cowley, there is no dallying with life), they determined to retire, and pass the remainder of their days in the country.

Notwithstanding all this care of Cicero, history informs us that Marcus proved a mere blockhead; and that Nature (who, it seems, was even with the son for her prodigality to the father) rendered him incapable of improving, by all the rules of eloquence, the precepts of philosophy, his own endeavours, and the most refined conversation in

Athens.

Natural historians observe (for whilst I am in the country I must fetch my allusions from thence) that only the male birds have voices; that their songs begin a little before breeding time, and end a little after.

Dr. Clark has observed that Homer is more perspicuous than any other author; but if he is so (which yet may be questioned), the perspicuity arises from his subject, and not from the language itself in which he writes.

The many letters which come to me from persons of the best sense of both sexes (for I may pronounce their characters from their way of writing) do not a little encourage me in the prosecution of this my undertaking.

It is this sense which furnishes the imagination with its ideas; so that by the pleasures of the imagination, or fancy (terms which I shall use promiscuously), I here mean such as arise from visible objects.

The stomach (crammed from every dish, a tomb of boiled and roast, and flesh and fish, where bile, and wind, and phlegm, and acid, jar, and all the man is one intestine war) remembers oft the schoolboy's simple fare, the temperate sleep, and spirits light as air.

William Penn was distinguished from his companions by wearing a blue sash of silk network (which, it seems, is still preserved by Mr. Kett, of Seething Hall, near Norwich), and by having in his hand a roll of parchment, on which was engrossed the confirmation of the treaty of purchase and amity.

Again, would your worship a moment suppose (it is a case that has happened, and may be again) that the visage or countenance had not a nose, pray who would, or who could, wear spectacles then? Upon this the dial-plate (if we may credit the fable) changed countenance with alarm.

To speak of nothing else, the arrival of the English in her father's dominions must have appeared (as indeed it turned out to be) a most portentous phenomenon.

Surely, in this age of invention, something may be struck out to obviate the necessity (if such necessity exists) of so tasking the human intellect.

I compassionate the unfortunates now (at this very moment, perhaps) screwed up perpendicularly in the seat of torture, having in the right hand a fresh-nibbed patent pen, dipped ever and anon into the ink-bottle, as if to hook up ideas, and under the outspread palm of the left hand a fair sheet of best Bath post (ready to receive thoughts yet unhatched), on which their eyes are riveted with a stare of disconsolate perplexity, infinitely touching to a feeling mind.

O the unspeakable relief (could such a machine be invented) of having only to grind an answer to one of one's dear five hundred

friends!

Have I not groaned under similar horrors, from the hour when I was first shut up (under lock and key, I believe) to indite a dutiful epistle to an honoured aunt ?

To such unhappy persons, then, I would fain offer a few hints (the fruit of long experience), which may prove serviceable in the hour of emergency.

If ever you should come to Modena (where, among other relics, you may see Tassoni's bucket), stop at a palace near the Reggio gate, dwelt in of old by one of the Donati.

My father and my uncle Toby (clever soul) were sitting by the fire with Dr. Slop; and Corporal Trim (a brave and honest fellow) was reading a sermon to them.

As the sermon contains many parentheses, and affords an opportunity also of showing you a sentence in brackets (you will observe that all the previous parentheses in this lesson are enclosed in crotchets), I shall insert part of it in the following paragraph:

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To have the fear of God before our eyes, and in our mutual dealings with each other, to govern our actions by the eternal measures of right and wrong: the first of these will comprehend the duties of religion; the second those of morality, which are so inseparably connected together, that you cannot divide these two tables, even in imagination (though the attempt is often made in practice), without breaking and mutually destroying them both. [Here my father observed that Dr. Slop was fast asleep]. I said the attempt is often made; and so it is; there being nothing more common than to see a man who has no sense at all of religion, and, indeed, has so much Lonesty as to pretend to none, who would take it as the bitterest front should you but hint at a suspicion of his moral character, or Imagine he was not conscientiously just and scrupulous to the utter

most mite.

I know the banker I deal with, or the physician I usually call in ["There is no need," cried Dr. Slop (waking) "to call in any physician in this case"], to be neither of them men of much religion. Experienced schoolmasters may quickly make a grammar of boys' natures, and reduce them all (saving some few exceptions) to certain general rules.

Ingenious boys, who are idle, think, with the hare in the fable, that, running with snails (so they count the rest of their school-fellows), they shall come soon enough to the post; though sleeping a good while before their starting.

VIII. THE DASH.

48. The Dash is a short straight line which occurs in reading, and which is placed between the sentences in such a manner as to be parallel the top or the bottom of the page.

49. The dash is sometimes used to express a sudden stop, or change in the subject.

50. The dash requires a pause sometimes as short as that of a comma, and sometimes one as long as, if not longer than, that of a period.

51. The dash is frequently used instead of crotchets or brackets, and a parenthesis is thus placed between two dashes. 52. The dash is sometimes used to precede something unexpected; as when a sentence beginning seriously ends humorously. 53. In the following examples, the dash is used to express a sudden stop, or change of the subject.

Examples.

If you will give me your attention, I will show you-but stop, I do not know that you wish to see.

Alas! that folly and falsehood should be so hard to grapple withbut he that hopes to make mankind the wiser for his labours, must not be soon tired.

"Please your honours," quoth Trim, "the inquisition is the vilest-" "Prithee, spare thy description, Trim; I hate the very name of it," said my father.

The fierce wolf prowls around thee-there he stands listening-not fearful, for he nothing fears.

The wild stag hears the falling waters' sound, and tremblingly flies forward-o'er his back he bends his stately horns-the noiseless ground his hurried feet impress not-and his track is lost amidst the tumult of the breeze, and the leaves falling from the rustling trees.

The wild horse thee approaches in his turn. His mane stands up erect his nostrils burn-he snorts-he pricks his ears and starts aside.

There was silence-not a word was said their meal was before

them-God had been thanked, and they began to eat.

They hear not-see not-know not-for their eyes are covered with thick mists-they will not see.

And ye like fading autumn leaves will fall; your throne but dustyour empire but a grave-your martial pomp a black funereal pallyour palace trampled by your meanest slave.

To-day is thine-improve to-day, nor trust to-morrow's distant ray.

For some time the struggle was most amusing-the fish pulling, and the bird screaming with all its might-the one attempting to fly, and the other to swim from its invisible enemy-the gander at one moment losing and the next regaining his centre of gravity.

54. The dash is sometimes to be read as a period, with the falling inflection of the voice.

Examples.

The favoured child of Nature, who combines in herself these united

perfections, may justly be considered as the masterpiece of creation— as the most perfect image of the Divinity here below.

Now launch the boat upon the wave-the wind is blowing off the shore-I will not live a cowering slave, in these polluted islands

more.

The wind is blowing off the shore, and out to sea the steamers flymy music is the dashing roar, my canopy the stainless sky-it bends above, so fair a blue, that heaven seems opening to my view.

He had stopped soon after beginning the tale-he had laid the frag. ment away among his papers, and had never looked at it again. The exaltation of his soul left him-he sunk down-and his misery went over him like a flood.

Mr. Playfair was too indulgent, in truth, and favourable to his friends-and made a kind of liberal allowance for the faults of all mankind-except only faults of baseness or of cruelty; against which he never failed to manifest the most open scorn and detestation.

Towards women he had the most chivalrous feelings of regard and attention, and was, beyond almost all men, acceptable and agreeable in their society-though without the least levity or pretension unbecom. ing his age or condition.

55. The dash is sometimes to be read like a comma, with the voice suspended.

Examples.

"I have always felt that I could meet death with composure; but I did not know," she said, with a tremulous voice, her lips quivering-"I did not know how hard a thing it would be to leave my children, till now that the hour is come."

And Babylon shall become-she that was the beauty of kingdoms, the glory of the pride of the Chaldeans-as the overthrow of Sodom and Gomorrah by the hand of God.

Our land-the first garden of liberty's tree-it has been, and shall yet be, the land of the free.

They shall find that the name which they have dared to proscribethat the name of Mac Gregor is a spell.

Delightful in his manners-inflexible in his principles-and generous in his affections, he had all that could charm in society, or attach in private.

The joys of life in hurried exile go-till hope's fair smile, and beauty's ray of light, are shrouded in the griefs and storms of night.

Day after day prepares the funeral shroud; the world is grey with age: the striking hour is but an echo of death's summons loud-the jarring of the dark grave's prison door. Into its deep abyss-devouring all-kings and the friends of kings alike must fall.

She made an effort to put on something like mourning for her son; and nothing could be more touching than this struggle between pious affection and utter poverty: a black ribbon or so a faded black handkerchief, and one or two more such humble attempts to express by outward signs that grief that passeth show.

LESSONS IN GEOMETRY.-V.

SIMPLE GEOMETRICAL THEOREMS.

BEFORE entering on the consideration of problems in geometry which will be found to be practically useful to all who are engaged in any mechanical art, it will be necessary for the learner to become acquainted with a few simple statements or facts in geometry, the truth of which is so clear and plain that they require but little, if any explanation. These are called theorems, or self-evident propositions, from the Greek Oewpnua (the-o-re-ma), literally a sight, or something which can be seen, in contradistinction to problems, or propositions which require something to be done in order to effect their solution. The word "problem" is derived from the Greek роßλnμа (pro-ble-ma), which is derived in its turn from πро (pro) before, and Barλw (bal-lo) to cast or throw, while the word "proposition is derived from the Latin pro, before, and pono, to place. Hence the meaning of the words "problem" and " position " is precisely the same, namely, something that is placed before you to be done or solved.

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B

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DBE is a right angle, the straight line B E being at right
angles to the straight line c D, and making the adjacent
angles D B E, E B C equal to one ano-
ther. The pupil will remember that the
measure of an angle is the extent of the
opening of the lines or legs of which the
angle is formed. Thus, the sum of the
openings of the two angles A B C, A B D,
or the sum of the openings of the three
angles CBA, A BE, E B D is equal to the sum
of the openings of the angles C B E, E B D.
Thus we learn that if any number of straight lines meet in a
point in another straight line on one side of it, the sum of the
angles which they make with this straight line and with each
other are equal to two right angles; and if any number of
straight lines meet in the same point on the other side of it, the
angles thus made are also equal to two right angles. Hence
the angles made by any number of lines meeting together in the
same point are together equal to four right angles.

Fig. 3.

As a familiar illustration of this, the spokes of a wheel may be taken, which radiate from the nave as a common centre. If a chalk line were drawn down the middle of each spoke, these lines would meet in the centre of the nave, and the angles formed by these lines at their point of meeting would be equal to four right angles.

4. Any angle drawn in a semicircle is a right angle.

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An angle drawn in a semicircle is one which has its top or vertex in the arc, while its legs pass through the extremities of the diameter at its points of contact with the arc. Thus, the angle A C B in the semicircle A C B is a right angle. The truth of this may be shown by cutting out a right-angled triangle and applying it to a semicircle. If large enough, it will be found that the legs of the right angle will pass through the ends of the diameter of the semiLet the straight line A B intersect the straight line C D in the circle, no matter at what point in the arc of the semicircle the point E. Now, by the intersection of

1. When one straight line intersects another straight line, the vertical or opposite angles are equal to one another.

these two straight lines, four angles are formed, namely, C E A, A E D, DE B, and B E C. Of these the vertical or opposite angles are equal, namely, CEA to D E B, and A E D to CE B.

Fig. 4.

B

vertex of the right angle may be placed.

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The truth of this may be shown in a very simple and practical manner by copying the figure on a piece of paper, and then cutting out the angles and placing them on each other, the greater on the greater and the less on the less. This mode of proof will frequently be found useful in similar cases.

Opposite angles are also called vertical angles, because the top or vertex of each angle is directly opposite to the vertex of the other.

2. When a straight line intersects two parallel straight lines, the alternate angles are equal.

E

Let the straight line E F intersect the parallel straight lines A B, C D, in the points G H. The angles AGH, G H D are alternate angles, and are equal to one another, and the angles CH G, HG B are also alternate and equal.

G

H

Fig. 2.

B

D

There are eight angles formed by the intersection of the straight lines A B, C D, E F, in Fig. 2. Of these the reader will find that there are two sets of four angles that are equal to one another-namely, A G E B G H = GHCDH F, and EG B AGH GHD=CH F. Let him demonstrate the truth of this practically by drawing the figure on paper, cutting out one of the greater angles and one of the less, and placing them on the remaining angles in each set of four.

3. The adjacent angles which are formed when one straight line stands on another straight line, are together equal to two right angles.

In Fig. 3 the adjacent angles A B C, A B D, which are formed by the straight line A B standing on the straight line C D, are equal to two right angles. The truth of this is evident when we consider that each of the angles C B E,

B

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In the triangle A B C in Fig. 5, of the three angles-A B C, B CA, CAB-ABC is manifestly the greatest; while of the three straight lines A B, B C, C A, which form its sides, AC is the greatest. A C, the greatest side, is opposite the greatest angle A B C; or, in other words, ▲ c, the greatest side, subtends the greatest angle A B C.

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D

Fig. 5.

A moment's reflection will show that the greatest angle of any triangle must have the greatest opening between the lines of which it is formed, and that the line which is opposite to or subtends the greatest opening, must of necessity be greatest of the three lines which subtend the three openings of the angles of the triangle.

6. If one side of a triangle be produced, the outer or exterior angle is equal to the two interior and opposite angles of the triangle. In the figure that accompanies the preceding theorem let the side A c of the triangle A B C be produced to D. The outer or exterior angle B C D is equal to the two interior and opposite angles C B A, B A C. For if at the point c in the straight line A D the straight line c E be drawn parallel to ▲ B, then the alternate angles E C B, C B A are equal to one another, and by Theorem 2, the angle D C E is equal to the angle C A B; but the angles D C E, E C B together make up the angle D C B, which is therefore equal to the angles C B A, B A C.

7. The three interior angles of every triangle are together equal to two right angles.

In Fig. 5 the angle B C D has been shown to be equal to the angles C BA, BAC; to each of these equals add the angle BCA. Now, by Theorem 3 the angles D C B, B C A are equal to two right angles, and C B A, B A C, A C B, the three interior angles of the triangle A B C, which are equal to these two angles, must therefore be equal to two right angles.

PROBLEMS IN PRACTICAL GEOMETRY.

PROBLEM I.-To bisect a given straight line—that is, to divide it into two equal parts.

Let A B (Fig. 6) be the straight line to be bisected. From the two extremities A and B, with a radius of any length greater than half of the line, describe or draw arcs of circles, intersecting or crossing each other at the point c, above the straight line AB, and at the point D, below it. Then, from the point of intersection c, draw a straight line to the point of intersection D; and the straight line A B will be bisected by the straight line C D, at the point E; that is, A B is divided into two equal parts, A E, E B, at the point E.

By this method of construction, a straight line may be divided into any number of equal parts, denoted by the series 2, 4, 8, 16, 32, 64, 128, etc.

It is not necessary in the above construction that the two arcs at D

A

E

Fig. 6.

B

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D

Let A F (Fig. 7) be the straight line to which the perpendicular is to be drawn, and в the point in it. From the point B, with any convenient radius, less than B A or B F, cut off, or measure off equal parts of the straight lines B A, B F-namely, в C, BE; and from the points C, E, with any radius greater than c B or E B, describe arcs of circles intersecting Fig. 7. each other at the point D. Then join D B, that is, draw a straight line from the point D to the point B, and B D will be perpendicular

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to A F.

B

E Y

PROBLEM III-To draw a perpendicular to a straight line from one of its extremities.

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Let A B (Fig. 8) be the straight line, and B one of its extremities, from which the perpendicular is to be drawn. Take any point c, at a convenient distance from B, and nearly over the middle of the straight line A B ; then with c as a centre, at the distance C B as radius, describe the arc D B E, so that it shall be greater than a semicircle; from the point D, draw through the point c, the straight line D C E, to meet the arc in the point E; and join E B, that is, draw a straight line from the point E to the point B, and B E will be perpendicular to A B, at the extremity of B, as required.

A

D

Fig. 8.

B

The demonstration of this proposition is founded on the fact that the angle contained in a semicircle is a right angle. This fact, indeed, is well known to intelligent workmen, who are accustomed to make use of the For the T square; for they try the accuracy of that instrument by this property of the circle. Thus, if in Fig. 9 A G C were an angle drawn by means of an F or T square, in order to test its accuracy, and consequently that of the instrument, they join any two points in the legs of the angle, say D C, by drawing the straight line DC; they bisect it in E by means of the arcs shown in the figure on either Fig. 9. side of the straight line CD, and drawn by the method explained in Problem I.; and then, with radius E C or E D, they describe the semicircle DGC; if the arc of this semicircle passes exactly

through the point G, the angle and the instrument are correct; if not, they are incorrect, and the instrument must be adjusted. PROBLEM IV.-To draw a perpendicular to a straight line from a point without it.

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Let A B (Fig. 10) be the straight line, and c the point from which the perpendicular is to be drawn. From the point c as a centre, with any radius sufficient to extend beyond the straight line A B, describe an arc of a circle D E, intersecting the straight line A B in the points D, E; then, from these points as centres, with any radius greater than half the straight line D E, describe arcs intersecting each other in the point F; then join c F; that is, draw a straight line from c to F, cutting A B in the point &; then c G is perpendicular to A B, and is drawn from the point c, as required. PROBLEM V.-To draw a perpendicular to a straight line at or near one of its extremities, from a point without it.

F

Fig. 10.

Let A G (Fig. 9) be the straight line, G one of its extremities, and c the point without it, from which the perpendicular is to be drawn. Take any point D in A G, and join D C; bisect it in E; and from the point E, as a centre, with radius E D or E C, describe the semicircle D G C; then join G c, and it will be perpendicular to a G. It is evident, from the remarks made on Problem III., that c & is perpendicular to a G, and it is drawn from the point c, as required.

Observe, that unless the point happens to be exactly in the vertical line above the point &, the semicircle will not pass exactly through G, but will pass through a point either nearer to or farther from the point A. In the latter case, the straight line A G must be produced till it meets the arc of the semicircle. This problem is considered as merely a case of the preceding problem, although the construction be different.

HISTORIC SKETCHES.-V.

THE RISING OF THE LABOURERS UNDER RICHARD II.

ON Whit Monday, 1382, Sir Simon Burley, who is called by one historian "a favourite of King Richard II.," and by another, "a Knight of the King's Household," rode into Gravesend, and seeing one of the townsmen, claimed him as his slave. There was great dissatisfaction and open murmuring among the people, with whom the man was a favourite, and they protested against his removal. The townsman himself loudly declared that he never was slave to any one, to Sir Simon or another, and seeing the sympathy the crowd had with him, he appealed to them for help. Sir Simon claimed the man as the son of one of his female slaves, called niefs, and disregarding the earnest entreaty of the crowd, would not abate his claim unless he were' paid three hundred pounds of silver-a price he well knew the friends of the bondman could not possibly raise. Some disorder ensuing, Sir Simon, who was attended by two serjeants of law and a following of armed men, pushed on through the crowd, and gave orders that the prisoner should be taken to Rochester Castle.

As soon as the great man's train had left, the awe inspired by its presence died away, and the people, whom the seizure of their fellow had taken completely by surprise, and had also deprived of their power to act, recovered their self-possession, and began to cry out with one voice, "Down with the tyrants! Let us go to Rochester! Let us join our brethren of Essex!"

The Essex men had already risen in arms, and were vowing vengeance on all the lords and owners of land, and especially against lawyers, whom they hated as the ministers of the law that crushed them. Norfolk, Suffolk, Cambridgeshire, and some of the other home counties, had been infected with the same spirit. In them the bubbles of rebellion were beginning to rise to the surface and to break, though as yet there was nothing like united action. The above-mentioned claim of Sir Simon Burley, made in spite of the ferment which was going on only on the opposite bank of the river, was the spark which fired the train of the Kentish men's anger.

Before time enough had elapsed to throw cold water on the fire, another and more serious offence had been given to the

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