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LESSONS IN BOTANY.-XX.

the circumstance of having imbricated æstivation and free

anthers, by the presence of an involucrum surrounding each SECTION XXXVIII.-COMPOSITÆ, OR COMPOSITE

flower; lastly, by the pendant and albuminous seed. FLOWERED PLANTS (continued).

The great family, Compositæ, is dispersed all over the globe ; We must not omit to mention, while discussing the various nevertheless, the number both of species and of individuals means taken advantage of boy Nature to promote the dissemina- rapidly diminishes towards either pole, and slightly towards the tion of Compositos, a very grotesque arrangement possessed by Equator. They chiefly inhabit temperate and hot regions, more certain species, in virtue of which animals are made the uncon especially tropical islands, and districts on the sea-coast of scious bearers of the precious vegetable charge. The bract tropical continents. America is richest in number of species. which we have already seen competent to assume so many Herbs belonging to this order grow in climates which are shapes, becomes in certain species of this natural order hooked, temperate and cold; shrubs in regions still hotter; and trees covering each involucre with hundreds of claw-like arms. Who in the hottest of all. Moreover, the latter are exclusively conhas not seen this curious provision on the burdock, though, fined to intertropical and antarctic islands. Tubulifloræ are perhaps, the utility of this singular appendage has not sug. numerous between the tropics, Liguliflora in the northern temgested itself? The use of this book, no doubt, is for the pur- perate region. Labiatiflor@ are rare out of America, where they pose of causing the torus to lay hold of the skins of animals or abound between the Equator and the Tropic of Capricoin. other passing objects.

Whatever may be the locality of any one species belonging to The Composito being a natural order which includes so large this order, it is rare that it can be naturalised elsewhere. In

174

175

Kinder

173

173. THE CORN CENTAUREA, OR CORN-FLOWER (CENTAUREA

CYANEA). 174. THE COMMON MARIGOLD (CALENDULA
OFFICINALIS), 175. THE DANDELION (LEONTODON
TARAXACUM, OE TARAXACUM OFFICINALE).

& number of species, some kind of subordinate classification this respect the Composite are peculiarly unbending ; neither becomes necessary. Botanists are by no means agreed as to care nor culture will generally suffice to effect a permanent the best method of accomplishing this. Perhaps the system of reconciliation between the transported plants and their now Decandolle and Endlicher is most generally convenient: accord. homes; to this, however, there are many exceptions. ing to which the order Compositæ is divided into three series ; The immense family of Compositæ furnishes mankind with first, Liguliflore, or strap-shaped flowers, from the Latin ligula, numerous useful products, some of which will now be rapidly 2 strap; second, Labiatiflore, or lip-shaped flowers, from the enumerated. The radiated Tubuliflore, regarded in the aggreLatin labium, a lip; third, Tubuliflore, or funnel-shaped gate, may be said to contain in the flower a bitter principle comflowers, from the Latin tubulus, a funnel or small tube. These bined with a resin or volatile oil; associated with these there is sab-families are divided into eight tribes, which are again frequently discoverable in the root a material something resem. divided into genera, and so each species is arrived at.

bling starch, and designated chemically by the specific name There are a few natural orders which, regarded in the tout inuline, because it is chiefly found in the elecampane (Inula). ensemble of their general characteristics, approach the Compo. According to the mutual proportions in which one or another site. The little family of Calyceraceæ presents a great analogy of these bodies may predominate, the various species become with them, both as regards the inflorescence and the structure endowed with different medicinal properties. Some are tonics, of individual flowers. It differs from Composite, however, in others excitants or stimulants, others are astringents. The the circumstances that the seed, instead of being erect at the great genus Artemisia, represented throughout all the world by base of the ovary, is suspended from the summit of the latter; different species, furnishes us with various bitter aromatics, the that the embryo is enclosed in a fleshy albumen; that the radicle properties of many of which have been celebrated from periods is superior ; that the style, always undivided, is terminated by a of very high antiquity. Two species, Artemisia Absinthium, and capitular stigma. Next come the Dipsaceæ, of which the greater Artemisia Pontica, are indigenous. Southernwood, or Arte. portion resemble the Composito by their inflorescence being that misia Abrotanum, originally from the East, is now cultivated in of a capitular involucrum; but which differ from the family in our gardens, and of world-wide reputation for its penetrating

of us.

INDICATIVE MOOD.

PRESENT.

IMPERFECT.

PERFECT.

odour. All these species of composite-flowered plants owe their merce, being employed as a substitute for, or an adulteration of, properties to the presence of a bitter principle, a peculiar acid, coffee. We should remark, however, that throughout Germany and a volatile oil. Perhaps the most valuable product of the and France the coffee-drinking public has become so accustomed Composite family is a volatile oil, acrid in some species, only to the flavour of coffee mixed with a certain amount of chicory, bitter in others. Pre-eminent in the list stands chamomile, that simple coffee is never by preference employed. Endiva useful in so many diseases. Arnica montana, a plant which (Cichorium Endivia), so much employed as a salad, is also one grows in Germany, Switzerland, and France, also owes its of the Chicoracec, etiolated, or bleached, by protecting it during medicinal qualities to the presence of a volatile oil.

growth from the direct action of air and light. Two varieties of The genus Helianthus, ir which the common sunflower is endive are known to gardeners; one with large oblong leaves, included, deserves especial notice for the products which it very slightly charged with the bitter principle; the other more yields. Helianthus tuberosus, the Jerusalem artichoke, is a decidedly bitter, and having leaves which are very much subperennial plant, indigenous to Brazil, though now cultivated in divided and crisped. various European countries. Its subterraneous stem produces The genus Lactuca, or lettuce, is a very important one belong. enormous tubercles, charged with inuline, and therefore very ing to the sub-tribe Chicoraceæ. All the members of this genus nutritive. Their odour is nauseous, but their taste agreeable ; are characterised by possessing a bitter acrid juice, and being sonsequently, after being well seasoned, they may be eaten by strongly odorous. All the lettuces contain wax, caoutchouc man. They resist the attack of frost, in which respect they or india-rubber, a resin, a bitter crystallisable matter, and a are different to most tubers, and consequently furnish good peculiar volatile principle. Most of the lettuce genus are medi. winter fodder for cattle. The Helianthus annuus, or sunflower, cinal, the predominant medical quality of each being determined is familiar to most of us. Its seeds afford, by expression, large by the preponderance of one principle. Even common garden quantities of a fixed oil admirably adapted for purposes of lettuce, in the condition in which we eat it as a salad, is known illumination and the soap manufacture. Wo shall now con- popularly to be endowed with soporific propertios ; but the clude this notice of radiated Tubuliflora by mentioning the narcotic energy is most strongly developed in the Lactuca virosa, sonchodendron, a tree fifty feet in height, and the largest of the a plant not uncommon in England. Compositæ. It is a native of Madagascar, in the deep valleys of which island it grows ; and although it does not furnish a product useful to man, it aids him in another way. When the

LESSONS IN LATIN.-XX. sonchodendron flowers, the natives know the best season has arrived for sowing their rice.

PARADIGM OF THE VERB SUM-COMPOUNDS OF SUM. The genus Cynara comprehends many species, of which one,

It will be convenient here to present the verb Esse, to be, in the common artichoke (Cynara Scolymus), is familiar to most full

. This verb is sometimes called an auxiliary verb, as by The part which we eat in this vegetable is the bracteal its aid (auxilium) parts of other verbs are formed. It is also involucre, or rather the fleshy base of each bract, and the com

called the substantive verb, as in its essence it denotes being or mon receptacle. The Italians are more expert in turning the

substance.
artichoke to account. They tie all the petioles together into
one mass, curve the plant at right angles, and surround it with
earth. In this manner, not only the capitulum, but the whole

Singular.
Singular.

Singular. apper portion of the plant, becomes etiolated, or bleached, and, Sum, I am.

Eram, I was. Fui, I have been. forms a sort of cabbage head, eaten as a salad by the Italians.

Es, thou art.

Eras, thou wast. Fuisti, thou hast been, Several individuals of the Carthamus tribe of Compositæ are

Est, he is.

Erat, he was. Fuit, he has been conspicuous on account of the colouring matter which they

Plural.
Plural.

Plural. yield. Of these the Carthamus tinctorius, or sa flower plant, is Samus, we are. Erämus, we were.

Fulmus, we have been. the most valuable. It is an annual, indigenous to India, but Estis, you are. Erātis, yo were. Fuistis, ye have been. now cultivated in various other parts of Asia, America, and Sunt, they are. Erant, they were. Fuerunt, they have beeile, Europe. Its florets contain two colouring principles, one of

FUTURE, which is much more soluble in water than the other. It is this Singular,

Singular,

Singular. latter, however, which the dyer seeks. Although rather it.sola- Fuěram, I had been. Ero, I shall be.

Fuero, I shall have beert, ble in water, it is easily extracted by alkaline leys, from which Fuðras, thou hadst been. Eris, thou shalt bo. Fueris, thou shalt hare ben. it admits of ready precipitation by the addition of an acid. Fuěrat, he had been. Erit, he shall be. Fuerit, he shall have been. The colouring principle thus obtained is denominated cartha

Plural.
Plural.

Plural. mine. The carthamine of Egypt and of Persia are most Fuerāmus, we had been. Erimus, we shall be. Fuerimus, we shall have beers esteemed; that of Spain follows next in order; that of France, Fuerant, they had been. Erunt, they shall be. Fuerint, they shall have been.

Erftis, ye shall be. Fueritis, ye shall have been Mexico, and Germany is of less value. Unfortunately, the tint communicated by safflower, although beautiful, is very fleeting.

SUBJUNCTIVE MOOD. Carthamus florets are frequently mingled with those of truo

IMPERFECT. saffron as an adulteration.

Sing. Sim, I may be.

Sing. Essem, I might be. The common marigold (Calendula officinalis) is cultivated in Sis, thou mayest be.

Esses, thou mightest be. gardens ; it contains a bitter mucilaginous substance, various

Sit, he may be.

Esset, he might be. salts, and a small quantity of volatile oil. It was formerly Plu, Simus, we may be.

Plu, Essemus, we might be. celebrated in medical practice, and is now again employed by

Sitis, ye may be.

Essetis, ye might be. the homeopathic practitioner (Fig. 174).

Sint, they may be.

Essent, they might be. The Liguliflore, or Chicoraceæ, contain a milky juice in their circulating vessels; also bitter, saline, resinous, and narcotic Sing. Fuerim, I may have been. Sing. Fuissem, I might have bem. principles. Their properties vary according to the predominance

Fueris, thott mayest have been..!

Fuisses, thou mightest harebem attained by one over the other of theso substances. The herb Plu. Fuerimus, we may have been.

Fuerit, he may have been,

Fuisset, he might have been, part of several of the Chicoracece, if cooked whilst young, before

Plu. Fuissēmus, the milky fluid has become completely formed, is an agreeable

Fueritis, ye may have been.

Fuissētis, ye might hare both article of food ; but the Chicorace are more celebrated in modi

Fuerint, they may have been,

Fuissent, they might have been cine than in dietetics. One of the most useful as well as the

IMPERATIVE MOOD. most common of Chicoracece is the dandelion (Taraxacum

Es or esto, be thou.

Este, or estöte, be ye. officinale, Fig. 175), a small perennial, having a wide distribution.

Esto, let him be.

Sunto, let them be, Not only is it found abundantly in the British Isles, but through.

INFINITIVE MOOD. out Europe, Asia, and Northern Africa. The chicory (Cichorium Present. Esse, to be.

Future. Fore, or futurum esse, to Intybus), remarkable amongst indigenous Compositæ for its blue Perfect. Fuisse, to have been.

be about to be flowers, is scarcely less common than the dandelion, and, perhaps, equally valuable as regards the result its yields. The root of Present. Eng, being (not used in good Latin

, but found in the compound

PARTICIPLES.
wild chicory is employed in medicine; that of garden chicory,
when dried and roasted, is the object of a considerable cor-

presens).
Futuro. Futurus -a, -um, about to be.

PLUPERPECT.

FUTURE PERFECT.

PRESENT.

PERFECT

PLUPERFECT.

we might have been

is.

vernor.

a

The verb esse is made up of parts of two separate verbs; first,

EXERCISE 75.-LATIN-ENGLISH. a verb of which es is the root; and secondly, of a verb, the stem 1. Quamdiu felix eris, multi tibi erunt amici. 2. Pugna fuit atrocig. of which is fu (compare fio in Latin, and puw, fu'-o, in Greek). sima, propterea quod utriusque exercitūs militos fortissimi fuerunt. From es (esum originally for sum) came the present, the imper-3. Ante belli initium in urbe fueramus. 4. Demosthenis ætate multi fect, and the first future tenses; from fuo came the perfect, the oratores magni et clari fuörunt, et antea fuerant, nec postea defuörunt. pluperfect, and second future tenses.

5. Hæc res non profuit nobis sed obfuit. · 6. Si quis virtutis compos The verb sum has neither gerund nor supine, and is in other erit, semper beatus erit. 7. Quamdiu sorte mea contentus ero, ero

felix. 8. Actio recta non erit, nisi recta fuerit voluntas. 9. Si probi respects defective, as appears from the paradigm just given.

fuerimus, non deerit hominum laus. 10. Attenti este, discipuli. 11. Sum takes before it certain prepositions, and is modified by Homines mortis memores sunto. 12. Contenti estote sorte vestrā! them in its meaning; thus, with ad, adsum, it means I am at 13. Mi fili, semper virtutis præceptorum memor esto! 14. Vir prudens or near ; with ab, absum, it means I am from, away from, absent; non solum præsentia curat, sed etiam præterita mente repetit, et futura with pro, prosum, it means I am for, that is, I aid or benefit. ex præteritis providet. 15. Boni bonis prodesse student. In prosum, the letter d is inserted to prevent the hiatus which

EXERCISE 76.-ENGLISH-LATIN. would be caused if two vowels came in succession; thus,

1. Our soldiers were very brave in the fight. 2. Why were our pro-(d)-es, pronounced prodes; also prodest, proderam, prodero, soldiers braver than yours in the fight? 3. So long as you are happy, prodessem.

friends will not fail you, 4. Friends fail the wretched, 5. Before the From the root mentioned above-namely, fu, fuo—come two beginning of the fight, I was in the city. 6. The brave will always be forms not so common as those given in the table-namely, useful to the brave. 7. My enemies injure me. 8. If you are partakers forem and fore; forem (-es, -et; -emus, -etis, -ent) is the imperfect of virtue, you will be happy. 9. So long as I am content with my lot, subjunctive, and signifies I might be ; corresponding to essem

I shall be happy. 20. 0 scholars, you ought to be attentive in school!

13. of the table; fore is the infinitive future, to be about to be; corre

11. They endeavour to be very brave. 12. Be brave, my son.

Prudent men foresee the future (pl.) from the past. sponding with the futurum esse of the table.

VOCABULARY.
VOCABULARY.

Et-et, and-and (and | Prius, adv., before. Quantum, how much. Absens, part., being Hodie, to-day.

Præsum, præfui, præ

-also), both.
Qualis, -e, of what kind Quantus,

-um, absent. Intersum, interfui, in. esse, I am before, I

In, towards.

(E. R. quality).

hove great (E.R.quanAbsum, abfui, abesse, teresse (E. R. inte- preside over, command Nescio, I know not.

Qua mente sis, of tity). rest), I am among, I Prosum, profui, prod. Nescius, -, -um, igI am absent.

what disposition you Scio, 4, I know (E. R. Adsum, adfui, adesse, am concerned, I take esse, I am for, I am

norant.

are, what your feeling science).. I am present. an interest or part in. useful, I do good to. Non sum nescius, I am

Tum, them. Arma, -orum, A., arms. Ita, so.

Pugna, -æ, f., a fight

aware of. Carolus, -i, m., Charles. Longe, far.

(E. R. pugilist). Concilio, 1, I reconcile, Magistrātus, -üs, m., Quamdiu, as long as, Observe that in indirect questions the dependent verb must unite. a magistrate or go. how long ?

be in the subjunctive (or dependent) mood; as, for example, Dum, conj., while.

Quum (pronounced narra mihi ubi fueris, tell me where you have been. Such a form Fera, -e, f., a wild beast Nisi, conj., unless.

cum), conj., when, is called an indirect question. The direct question would stand (E. R. fierce). Oratio, -önis, f., from the time when.

thus—abi fuisti? narra mihi, where hast thou been? tell me. In Poris, adv., out of doors. speech (E. R. orator). Ubi, adv., where, when the latter case the question is direct, and the verb, as not being Heri, yesterday. Peregre, abroad.

Ut, as.

dependent, is in the indicative mood; but pat narra mihi first, Observe that these compounds of sum require their object to and then your question is implied rather than stated; it is, be in the dative case; as, prodest MIHI, he does good TO ME, or therefore, an indirect question. In both direct and indirect queshe benefits ME.

tions the English is in the indicative; consequently, in putting EXERCISE 73.-LATIN-ENGLISH.

the dependent verb into English, you must in English use the 1. Deus omnibus locis adest. 2. Parvi pretii (of little arail), sunt indicative mood; but in putting the dependent verb into Latin, arma foris, nisi est consilium domi. 3. Contemnuntur ii qui nec sibi you must in Latin use the subjunctive mood. Compare what nec alteri prosunt, 4. Ut magistratibus leges, ita populo præsunt is said of the sequence of tenses, and similar and dissimilar magistratus. 5. Ratio et oratio conciliat inter se homines, neque ulla tenses, in the last lesson. re longius absumus a naturā ferārum. 6. Ego sum lætus, tu es tristis. 7. Si sorte vestri contenti estis, beati estis. 8. Dum nos in scholā

EXERCISE 77.-LATIN-ENGLISH. erämus, sorores nostræ in horto erant. 9. Quum Carolus heri domi 1. Non sum nescius quã mente tu in nos sis. 2. Scio qui mente tu nostræ erat, ego peregre eram. 10. Quamdiu tu et irater tuus domi in nos semper fueris. 3. Non sum nescius qui mente tu et prius in nostru eratis, tu lætus eras, sed frater tuus erat tristis. 11. Quamdiu nos fueris et nunc sis. 4. Non eram nescius quä mente tu in nos esges. tu aběras, ego eram tristis. 12. Cur heri in scholā non faisti? 13. Quia 5. Scio quam sint incerti animi hominum. 6. Cogita quam brevis sit cum patre pěregre fui. 14. Quamdiu tu et pater tuus domo abfuistis? vita! 7. Qualis sit animus, ipse animus nescit. 8. Cogita quantum 15. Sex menses abfuimus. 16. Cur milites nostri pugnæ non inter. nobis bona exempla prosint. 9. Incertus sum ubi frater meus nunc fuērunt ? 17. Quia longius abfuērunt, 18. Ubi heri fueras quum

sit. 10. Incertus sum ubi amicus meus et fuerit et nunc sit. 11. In. domi tuæ eram ?

certus eram ubi heri esses. 12. Narra nobis ubi heri fueritis. EXERCISE 74.-ENGLISH-LATIN.

EXERCISE 78.-ENGLISH-LATIN. 1. I am useful to thee. 2. Thou art useful to me. 3. The boys are 1. Tell me where thou art. 2. Tell me where thy father and mother not useful to (their) mothers. 4. Why are the girls not useful to 3. I know not where my sister is. 4. Dost thou know how much (their) fathers? 5. When thou wast absent, I was sad. 6. How long good boys do good (prosum) to their parents ? 5. I know where my has your father been absent? 7. Charles took part in the fight. 8.

6. My son, where art thou ? 7. I knew where my son was. Wast thou yesterday at my house ? 9. I shall be at thy house to-day. 8. I am uncertain where the enemies are. 9. Is the general ignorant 10. Unless thou art happy at home, thou art not joyful abroad. where the army is? 10. I know of what mind thou art toward the VOCABULARY.

king. Actio,-ōnis, f.,an action, Exercitus, -ūs, m., an Pretium, -i, n., a redoing (E. R. action). army.

ward (E. R. price,

LESSONS IN GEOMETRY.-XX. Ætas, ætatis, f., age, « Inimicus, -i, an enemy precious). generation. (E. R. inimical). Propterea, on account of

THERE is a curious connection between the proportions of the Amicus, -i, friend. Initium, -1., 1., a begin- Provideo, 3, I see before and that is, that the length of the side of the heptagon is equal

sides of a hexagon and a heptagon inscribed in the same circle, Antea, adv., before. ning (E. R. initial). hand, foresee (E. R. Atrox, -ocis, frightful Mens, mentis, f., mind provide).

to the perpendicular let fall from the centre of the circle on any (E. R, atrocious). (E. R. mental). Prudens, -tis, prudent, side of the hexagon. This may be seen from the following Attentus, -a, -um, at- Nunquam, never. Quod, because.

problem tentive,

Nisi, conj., unless, Rectus, -a, -um, right, PROBLEM LII.-To inscribe a heptagon in a given circle. Compos,-otis, partaking not.

Repěto, 3, I seek again,

Let A C (Fig. 74) be the given circle in which it is required to of, endued with. Obsum, obfui, obesse,

I repeat. Desum, defui, deesse, I am in the way of, I Sed, but.

describe a heptagon. Draw any diameter A K, passing through I am doron, I fail. oppose, I injure. Solum, alone.

L, the centre of the given circle A CE; and from one of its Demosthenes, -is, m., Postěa, afterwards. Studeo, 2, I endeavour. extremities A as centre, with the distance A L, describe the arc

Demosthenes, the cele- Præteritus, -2, -um, Voluntas, -ātis, f., will BL M, cutting the circumference of the circle ACE in the brated Grecian orator. past,

(E. R. voluntary.) points B and M. Join B M, cutting A K at right angles in the

are.

son is.

H

K

к.

N

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point n. From the point B as centre, with the distance B N, straight lines K M, L N, parallel to A o or B P, and along ku draw arcs cutting the circumference of the circle A CE set off K Q, equal to A B, and along L N set off L R, also equal in c and . Join C B, B H. These straight lines are sides to A B. Then from the points Q and R as centres, with a of a heptagon inscribed in the circle A C E, and the heptagon radius equal to A B, draw arcs cutting the perpendiculars a 0,

itself may be completed by applying B P, in s and T. Join Q 8, 8 T,
the compasses, with an opening equal T R. The figure A BL RTSQK
to B N or N M, round the circum- is an octagon, and it is described
ference of the circle A CE, as may on the given straight line A B as
be needful. The triangle A B L is an required.
equilateral triangle (see Problem Next, let A B (Fig. 77) be the
XVII., Vol. I., page 209), and A B is given straight line on which it is
the side of a hexagon that may be required to construct an octagon.
inscribed in the circle AC E, in which Produce A B indefinitely both
the heptagon B C D E F G H has already ways to x and y, and through
been inscribed.

the points A and B draw the
PROBLEM LIII.—To construct a straight lines A O, BP, perpen-
Fig. 74.

heptagon on any given straight line. dicular to A B or xy, and set off Let A B (Fig. 75) be the giren straight line on which it is along A o and B P the straight

Fig. 77. required to construct a heptagon. Produce A B indefinitely both lines A C, B D, each equal to ways to x and y. Bisect A B in c, and again bisect C Bin D. From A B. Join A D, B C, and produce them indefinitely to and B along the straight line B y set off B E equal to five times B D, respectively; and along D Q, CR, set off D E, C F, each equal to and from A along the straight line A F set off x equal to B E, A B. Through E and draw the straight lines EG, F !, meeting or five times B D. Then from the points A and B as centres, xy in G and u; and along E G, FH, set off E L and F K, each equal with the distances A E, B F respectively, describe the arcs E G, to A B. Through x draw k parallel to A Q, and cutting B Pin F G, cutting one another in the point G; and from G as centre, M; and through I draw L n parallel to B R, and cutting A o in s.

with a radius Join A K, B L, EM, MN, NF. The figure A B LEMN FK is an
equal to A B, de- octagon, and it is described on the given straight line A B, as
scribe the arc HK, required. If it be required to cut off the corners of a square
cutting the arcs piece of wood or pasteboard as A B C D in Fig. 78, so as to form
F G, E G, in the an octagon, first draw the diagonals a C,
points and K. BD, intersecting each other at right
Join G H and GK, angles in E, and then from the points
and bisect them A, B, C, D in succession, with the distances
respectively in the A E, B E, C E, D Е, describe the arcs
points, L, M; and F E G, H E K, L E M, and n E o, having
join A M, BL, their terminations in the sides of the

intersecting each square. Then join 0 H, G L, K N, and XF

other in the point MF. The figure M F O H G L K N is an Fig. 75.

N, which is the octagon.

centre of the circle PROBLEM LV.-To inscribe a nonacircumscribing the required heptagon. From the centre n, at gon in a given circle.

Fig. 78. the distance n A or N B, describe the circle A B K G H. Biseot Let A B C (Fig. 79) be the given circle the arcs A H, B K, in the points O, P, and join 2 0,0 A, B P, P K. in which it is required to inscribe a nonagon. Draw any diameter, The figure A BPKG H o is a heptagon, and it is described on C E passing through the centre d of the circle A B C, and produce the given straight line A B, as required.

it indefinitely towards F. From the point E as centre, with the PROBLEM LIV.-To construct an octagon on a given straight distance E D, describe the arc A D B, cutting the circumference of line.

the circle A B C in the points A and B. Join A B, and pro As it has been remarked in a former lesson (see page 192), it duce it indefinitely both ways towards G and y, and let it is easy to inscribe a hexagon in a given circle when we can place cut o F at right angles in the point k. Then from I an equilateral triangle within it, as the process is merely to as centre, with a radius equal to D Е, describe the semicircle bisect the arcs intercepted between the ends of the sides of the Lan, having its terminations L, n, in the straight line G ; triangle, and to form the hexagon by joining the six points thus and from L and N as centres, with the radi L K, N K respeo obtained in the circumference of the circle. By a similar pro- tively, describe the arcs KO, K P, meeting the semicircle LNS cess of bisection, an octagon may be inscribed in a given circle in the points o, P. Join DO, DP, cutting the circumferenes of when we have once placed a square within it; while the bisec- the circle A B C in the points Q, R. Join A Q, R, R B. These tion of the arcs intercepted between the ends of the sides of a three straight lines are the sides of a nonagon inscribed in the pentagon and hexagon will similarly produce a decagon and a circle A B C, which may be

dodecagon. There are, how. completed by following the
ever, one or two processes of same process with the arcs
constructing an octagon on a A C, C B, or, applying the
given straight line which we compasses round the cir-
give here, as they may be of cumference of the circle
use to the learner in cutting with an opening equal to
out an octagon in timber, or an A Q, R, or R B, and join.
octagonal flower-bed in turf. ing the points thus ob-
First, let A B (Fig. 76) be the tained. The straight line
given straight line on which it A B is manifestly the side
is required to construct an octa- of an equilateral triangle

gon. Produce A B indefinitely inscribed in the circle A BC,
Fig. 76.
both ways to x and y, and at the and the process which has

Fig. 79. points A and B draw the straight been gone through is simply lines A 0, B P, perpendicular to A B or x Y. From A as centre, the trisection of the arc A B, or, what is vistually the same thing, with the distance A B, describe the arc BCN, cutting A o in c, the trisection of the angle A D B. and xy in d; and from B, as centre, with the distance B A, PROBLEM LVI.—To construct a nonagon on any given straight describe the arc A E F, cutting B p in E, and x y in F. Join line. C D, E F, and bisect them in G and u respectively. Join a G, Let A B (Fig. 80) be the given straight line on which it is BH, and produce AG to meet the arc BCD in K, and B H to required to construot & nonagon. Produce A B indefinitely meet the arc A EP in L. Through the points K, L draw the both ways to x and y, and on the straight line f, with

M

R

E

line A B.

the points A and B as centres, and the radii A B and BA the point M al centre, with the distance M A or M B, describe respectively, describe the semicircles A C D, B C E, intersecting the circle A B F. This circle passes through A, B, F, G, H, the each other in the point c. Bisect A B in F, and through the extremities of three sides of the required undecagon that have point c draw F G perpendicular to A B or x y. Next trisect the been already determined, and the vertices of the remaining aro A c in the points 1 and K, and the arc B c in the points angles of the polygon will be found in its circumference. To L and m; and from the point a, through the points M, L, and o in the arc B C, draw the straight lines a N, AO, AP of indefinite length; and from B, through the points H, K, and c, in the arc A.C, draw the straight lines B 8, B R, and B Q, also of indefinite length. From the points A and B draw the straight lines A T, Bu to the points, t, u, in which the straight lines B S, AN cut the semicircles BCE, AC D; bisect AT, Bu in the points V and w respectively, and through the points v and w draw the perpendicular lines V 1, w z of indefinite length, intersecting each other and the straight line F g in the point a. This point is the centre of a circle circumscribing the required nonagon. From the point a as centre, with the distance a A or a B, describe the circle Ad B. This circle passes through the extremi. ties of the given straight line A B, the points T and u in which the straight lines B s, A N respectively intersect the semicircles BC E, A C D, and the points c and e in which the straight lines BQ, WZ and AP, v I intersect each other: it also cats the

Fig. 81. straight lines B R, F G, and A o in the points b, d, and f. Join T b, b c, cd, de, ef, and f u: the figure Arb c d esu B is a determine them, with an opening of the compasses equal to A B, nonagon, and it is described, as required, upon the given straight set off from a, along the arc a G, the arcs À N, NO, O P, and

along th3 arc B H, from B, set off the arcs B Q, Q R, RS. Join The construction of the uneven-sided polygons, the heptagon the chords A N, NO, O P, PG, BQ, Q R, R 8, s H. The figure and nonagon, by the aid of the ruler and compasses, have been A B QE SHFGPON is an undecagon, and it is described on given to show the learner that there is no regular polygon of any the given straight line A B, as required. number of sides that could not be constructed without having The reader will have noticed, doubtless, that the method of recourse to the measurement of the angle of the polygon or the constructing an undecagon on a given straight line by a purely angle at the centre of the circumscribing circle. The construction geometrical process, as given above, is similar in all essential of the decagon and dodecagon on any given line by the ruler and details to the process used for constructing a heptagon on a compasses alone we do not give, because either figure may be con given straight line, and it is based in both cases on the structed by learners, if they will exercise a little thought, and it numerical relation of the straight line on which either is to be

will afford them two constructed, to the sides of an isosceles triangle whose vertex is useful exercises to the apex of the polygon, and whose base is the given straight do so.

We shall line. In the case of the heptagon, the proportion of the base to therefore conclude the sides of the isosceles, whose vertex is the apex of the polygon, our problems on the is as 1 to 21 or 2:25; and to construct a heptagon on any given construction of the straight line, we have only to produce it indefinitely both ways, regular polygons and find points on either side of each extremity at a distance with the method of equal to 14 of the given line, or to bisect the given line and set constructing an un- off on either side of the perpendicular section straight lines decagon or eleven- equal to 18 of the given line. In the case of the undecagon, sided figure on a the proportion of the base to the side of the isosceles triangle, given straight line, whose apex is the vertex of the polygon, is as 1 to 3 or 3.5; and and then bring our to construct an undecagon on any given straight line, we have Lessons on Geome- only to produce the given straight line indefinitely both waya, try to an end with and set off from either extremity lines equal to 21 of the given a brief description straight line, or to bisect the given line, and from the point of

of the methods used bisection to set off on either side of it, along the given line proFig. 80.

for drawing the el. duced indefinitely, lines equal to 3 times the given straight line.

lipse, parabola, and We have added these remarks on the geometrical constructions hyperbola, curves made by the section of a right cone in parti- that we have given of the heptagon and undecagon, in the hope cular directions; the mode of tracing a spiral; and one or two that they may give the student a clue to other geometrical conother things, such as the connection of two curves by a straight structions of a similar character. We also recommend to his line, etc., which may be of practical use to our students. notice the geometrical construction of the nonagon, based on the

PROBLEM LVII.To construct an undecagon on any given preliminary construction of an equilateral triangle on the given straight line.

straight line on which it is required to construct the nonagon, Let A B (Fig. 81) be the given straight line on which it is re- and the trisection of the angles on either side of the base, or quired to construct an undecagon. First bisect A B in c, and pro- the arcs that are described opposite to them by drawing semi. dace A B indefinitely both ways to x and y. Then along c x set circles from either extremity of the base as centres, with a radius off a line, C D, equal to three times A B, or six times c A, and equal to the base. along C y set off a line, C E, equal to CD. From the point A as In drawing figures to exhibit the methods of constructing the centre, with the distance A E, describe the arc E t, and from the different polygons, from the pentagon to the undecagon, that point B as centre, with the distance B D, describe the arc D 2, have been given in detail in this and preceding lessons, the and let the arcs E T, D Z intersect each other in the point F. student is advised, for the sake of accuracy, to make them on This point is the apex of the undecagon, the straight line A B on a large scale ; as, if he attempt to construct his figures in the which it is constructed being considered as its base. From the limited space in which are drawn the figures that are used to point F as centre, with a radius equal to A B, draw small arcs illustrate our Lessons in Geometry, he may fail to complete cutting the larger arcs D Z, E T in G and H, and draw the chords | them to his satisfaction, in consequence of not being able to FG, FH. Join CF: the straight line drawn from c, through F, is draw the straight lines and arcs, of which the figures are com. perpendicular to A B, and the centre of the circle circumscribing posed, of suitable fineness, and to subdivide the arcs, whenever the required undecagon will be in CF. To find the centre, bisect it is necessary to do so, with sufficient accuracy. In all cases, FG, F I in the points K and L, and join A L, B K. The straight for the sake of good practice, the straight line on which a polylines A L, B K intersect each other and the straight line C F in the gon is to be constructed, should never be taken less than an inch point , which is the centre of the circumscribing circle. From | in length.

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