LESSONS IN PENMANSHIP.-XXXI. LEGAL HANDWRITING.-I. We now introduce to the notice of our readers handwriting of a very different description to any that has yet been set before them, namely, an illustration of the "engrossing hand" used by law writers and law stationers in writing, or, to use the technical expression, engrossing deeds and legal documents of various kinds. Amongst lawyers the term engrossing means the making of a fair copy of any document on paper or parchment in clear and distinct characters, or, in other words, to transcribe in a legible manner the rough draft of any deed that has been prepared for transcription by a lawyer or barrister. This knowledge will help us to ascertain the meaning of the word engross. It is derived, as a little examination is sufficient to show, from the French preposition en, in, and the adjective gros, large. The expression to engross, in its literal acceptation, means to make Whereas writing, which is slanting and angular, while the former is stiff and upright. The characters in which the words that form the remainder of our specimen of legal engrossing-hand are written, are merely modifications of the ordinary German text, or, which is pretty much the same thing, our ordinary round-hand letters written in a stiff, upright manner, and in such a way as to impart to them the general characteristics of the letters used in German text-hand. A little practice will enable the reader to write a very creditable engrossing-hand when he has once found out the way in which the pen ought to be held, and detected the method that is followed in the formation of the different letters. It is clear, in the first place, that the pen cannot possibly be held in the ordinary way, with the end pointing over the right shoulder in a direction from right to left, as many of the perpendicular and horizontal thick strokes that appear in the writing could not be made when the pen is held in such a position. The pen (and for engrossing a quill should be used in preference to a the said sums of mony aud Stork are now standing in the names of the said Trustees in the Books at the Bank of Euglaud, and the same suus and all intoust which has arrived but антиб tacucou siure the Testator's death belongs to the parties of the third and fourth parts in cqual shares and proportious. SPECIMEN OF LEGAL ENGROSSING HAND. large, and therefore distinct, as the larger in reason are the characters in which a document is written, the more distinct and legible they are. Hence the word engross, which, in its primary sense, merely means to make large, has taken a secondary meaning, namely, to copy writing in large or distinct characters; and we can also trace the force of the simple and original meaning in a third signification that has been assigned to the word, when we speak of a person who is occupied in some absorbing pursuit as being wholly engrossed in his occupation-his employment, be it what it may, having taken such a hold on him, and assumed such large proportions in his mind, that he has little or no room there, if we may be permitted to use the expression, for other thoughts. In the specimen of legal engrossing-hand before us, we may trace the very source from whence it has sprung by an examination of the characters that are used in it. Of the letters that compose the first word of the first line, Whereas, there can be no doubt whatever, as they are simply characters written boldly and clearly in what is called German text, so termed from its close resemblance to the characters in which the majority of German works are printed. It must be borne in mind that German text in no way resembles the ordinary German hand steel pen) is held pointing over the knuckle of the forefinger of the right hand in a direction bearing from left to right, while the nib is placed at such an angle with the paper that a thick perpendicular or horizontal stroke may be readily made, the former by turning the pen a little to the left and drawing it downwards in a straight line, and the latter by turning the pen a little to the right and drawing it in a transverse direction from left to right. The remainder of the strokes that form the rest of the letters in our specimen are formed, for the most part, by moving the pen downwards in a semi-circular motion from left to right. The kind of stroke that is meant will be recognised on looking at the letters m and n, which are formed entirely by a repetition of the stroke, thrice for the letter m, and twice for the letter n. The small letters, C and e, which may be seen together in since, the second word in the sixth line, present the most striking peculiarities in engrossing-hand: the former is a short, thick, perpendicular stroke, crossed at the top by a short horizontal line slightly turned upwards towards the finish; while thee is the stroke of which the letters m and n are formed, with a stroke rather longer and turned up more at its completion than that with which the letter C is finished. LESSONS IN BOTANY.-XX. FLOWERED PLANTS (continued). We must not omit to mention, while discussing the various means taken advantage of by Nature to promote the dissemination of Composito, a very grotesque arrangement possessed by certain species, in virtue of which animals are made the unconscious bearers of the precious vegetable charge. The bract which we have already seen competent to assume so many shapes, becomes in certain species of this natural order hooked, covering each involucre with hundreds of claw-like arms. Who has not seen this curious provision on the burdock, though, perhaps, the utility of this singular appendage has not suggested itself? The use of this hook, no doubt, is for the purpose of causing the torus to lay hold of the skins of animals or other passing objects. The Composite being a natural order which includes so large the circumstance of having imbricated aestivation and free anthers, by the presence of an involucrum surrounding each flower; lastly, by the pendant and albuminous seed. The great family, Compositæ, is dispersed all over the globe; nevertheless, the number both of species and of individuals rapidly diminishes towards either pole, and slightly towards the Equator. They chiefly inhabit temperate and hot regions, more especially tropical islands, and districts on the sea-coast of tropical continents. America is richest in number of species. Herbs belonging to this order grow in climates which are temperate and cold; shrubs in regions still hotter; and trees in the hottest of all. Moreover, the latter are exclusively confined to intertropical and antarctic islands. Tubuliflora are numerous between the tropics, Liguliflora in the northern temperate region. Labiatiflora are rare out of America, where they abound between the Equator and the Tropic of Capricorn. Whatever may be the locality of any one species belonging to this order, it is rare that it can be naturalised elsewhere. In 173 174 175 173. THE CORN CENTAUREA, OR CORN-FLOWER (CENTAUREA CYANEA). 174. THE COMMON MARIGOLD (CALENDULA OFFICINALIS). 175. THE DANDELION (LEONTODON TARAXACUM, OR TARAXACUM OFFICINALE). a number of species, some kind of subordinate classification becomes necessary. Botanists are by no means agreed as to the best method of accomplishing this. Perhaps the system of Decandolle and Endlicher is most generally convenient: according to which the order Composita is divided into three series; first, Liguliflora, or strap-shaped flowers, from the Latin ligula, 2 strap; second, Labiatiflora, or lip-shaped flowers, from the Latin labium, a lip; third, Tubuliflora, or funnel-shaped flowers, from the Latin tubulus, a funnel or small tube. These sub-families are divided into eight tribes, which are again divided into genera, and so each species is arrived at. There are a few natural orders which, regarded in the tout ensemble of their general characteristics, approach the Composite. The little family of Calyceraceae presents a great analogy with them, both as regards the inflorescence and the structure of individual flowers. It differs from Composite, however, in the circumstances that the seed, instead of being erect at the base of the ovary, is suspended from the summit of the latter; that the embryo is enclosed in a fleshy albumen; that the radicle is superior; that the style, always undivided, is terminated by a capitular stigma. Next come the Dipsacea, of which the greater portion resemble the Composite by their inflorescence being that of a capitular involucrum; but which differ from the family in this respect the Composite are peculiarly unbending; neither care nor culture will generally suffice to effect a permanent reconciliation between the transported plants and their new homes; to this, however, there are many exceptions. The immense family of Composite furnishes mankind with numerous useful products, some of which will now be rapidly enumerated. The radiated Tubuliflora, regarded in the aggregate, may be said to contain in the flower a bitter principle combined with a resin or volatile oil; associated with these there is frequently discoverable in the root a material something resembling starch, and designated chemically by the specific name inuline, because it is chiefly found in the elecampane (Inula). According to the mutual proportions in which one or another of these bodies may predominate, the various species become endowed with different medicinal properties. Some are tonics, others excitants or stimulants, others are astringents. The great genus Artemisia, represented throughout all the world by different species, furnishes us with various bitter aromatics, the properties of many of which have been celebrated from periods of very high antiquity. Two species, Artemisia Absinthium, and Artemisia Pontica, are indigenous. Southernwood, or Artemisia Abrotanum, originally from the East, is now cultivated in our gardens, and of world-wide reputation for its penetrating odour. All these species of composite-flowered plants owe their properties to the presence of a bitter principle, a peculiar acid, and a volatile oil. Perhaps the most valuable product of the Composite family is a volatile oil, acrid in some species, only bitter in others. Pre-eminent in the list stands chamomile, useful in so many diseases. Arnica montana, a plant which grows in Germany, Switzerland, and France, also owes its medicinal qualities to the presence of a volatile oil. The genus Helianthus, in which the common sunflower is included, deserves especial notice for the products which it yields. Helianthus tuberosus, the Jerusalem artichoke, is a perennial plant, indigenous to Brazil, though now cultivated in various European countries. Its subterraneous stem produces enormous tubercles, charged with inuline, and therefore very nutritive. Their odour is nauseous, but their taste agreeable; consequently, after being well seasoned, they may be eaten by man. They resist the attack of frost, in which respect they are different to most tubers, and consequently furnish good winter fodder for cattle. The Helianthus annuus, or sunflower, is familiar to most of us. Its seeds afford, by expression, large quantities of a fixed oil admirably adapted for purposes of illumination and the soap manufacture. We shall now conclude this notice of radiated Tubuliflora by mentioning the sonchodendron, a tree fifty feet in height, and the largest of the Composite. 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 sonchodendron flowers, the natives know the best season has arrived for sowing their rice. The genus Cynara comprehends many species, of which one, the common artichoke (Cynara Scolymus), is familiar to most of us. The part which we eat in this vegetable is the bracteal involucre, or rather the fleshy base of each bract, and the common receptacle. The Italians are more expert in turning the 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 upper portion of the plant, becomes etiolated, or bleached, and forms a sort of cabbage head, eaten as a salad by the Italians. Several individuals of the Carthamus tribe of Composite are conspicuous on account of the colouring matter which they yield. Of these the Carthamus tinctorius, or safflower plant, is the most valuable. It is an annual, indigenous to India, but now cultivated in various other parts of Asia, America, and Europe. Its florets contain two colouring principles, one of which is much more soluble in water than the other. It is this latter, however, which the dyer seeks. Although rather insoluble in water, it is easily extracted by alkaline leys, from which it admits of ready precipitation by the addition of an acid. The colouring principle thus obtained is denominated carthamine. The carthamine of Egypt and of Persia are most esteemed; that of Spain follows next in order; that of France, Mexico, and Germany is of less value. Unfortunately, the tint communicated by safflower, although beautiful, is very fleeting. Carthamus florets are frequently mingled with those of true saffron as an adulteration. The common marigold (Calendula officinalis) is cultivated in gardens; it contains a bitter mucilaginous substance, various salts, and a small quantity of volatile oil. It was formerly celebrated in medical practice, and is now again employed by the homœopathic practitioner (Fig. 174). merce, being employed as a substitute for, or an adulteration of, coffee. We should remark, however, that throughout Germany and France the coffee-drinking public has become so accustomed to the flavour of coffee mixed with a certain amount of chicory, that simple coffee is never by preference employed. Endive (Cichorium Endivia), so much employed as a salad, is also one of the Chicoracea, etiolated, or bleached, by protecting it during growth from the direct action of air and light. Two varieties of endive are known to gardeners; one with large oblong leaves, very slightly charged with the bitter principle; the other more decidedly bitter, and having leaves which are very much subdivided and crisped. The genus Lactuca, or lettuce, is a very important one belong. ing to the sub-tribe Chicoracea. All the members of this genus are characterised by possessing a bitter acrid juice, and being strongly odorous. All the lettuces contain wax, caoutchouc or india-rubber, a resin, a bitter crystallisable matter, and a peculiar volatile principle. Most of the lettuce genus are medicinal, the predominant medical quality of each being determined by the preponderance of one principle. Even common garden lettuce, in the condition in which we eat it as a salad, is known popularly to be endowed with soporific properties; but the narcotic energy is most strongly developed in the Lactuca virosa, a plant not uncommon in England. Sing. Plu. Sis, thou mayest be. Sint, they may be. PERFECT. INDICATIVE MOOD. IMPERFECT. Eram, I was. Plural. Eramus, we were. FUTURE. Singular. Ero, I shall be. Plural. Erimus, we shall be. Erunt, they shall be. PERFECT. Singular. Fui, I have been. Plural. Fulmus, we have been. Fuerunt, they have been.. FUTURE PERFECT. Singular. Fuero, I shall have bem. Fuerimus, we shall have been SUBJUNCTIVE MOOD. Fuerim, I may have been. IMPERFECT. Sing. Essem, I might be. Plu. Essemus, we might be. Essetis, ye might be. PLUPERFECT. Sing. Fuissem, I might have been, Fuisses, thou mightest have bem Fuissetis, ye might have been. IMPERATIVE MOOD. The Liguliflora, or Chicoracea, contain a milky juice in their circulating vessels; also bitter, saline, resinous, and narcotic principles. Their properties vary according to the predominance attained by one over the other of these substances. The herb part of several of the Chicoraceæ, if cooked whilst young, before the milky fluid has become completely formed, is an agreeable article of food; but the Chicorace are more celebrated in medicine than in dietetics. One of the most useful as well as the most common of Chicoraceae is the dandelion (Taraxacum officinale, Fig. 175), a small perennial, having a wide distribution. Not only is it found abundantly in the British Isles, but throughout Europe, Asia, and Northern Africa. The chicory (Cichorium Intybus), remarkable amongst indigenous Composite for its blue flowers, is scarcely less common than the dandelion, and, perhaps, equally valuable as regards the result its yields. The root of Present. Ens, being (not used in good Latin, but found in the compound The verb esse is made up of parts of two separate verbs; first, a verb of which es is the root; and secondly, of a verb, the stem of which is fu (compare fio in Latin, and pvw, fu'-o, in Greek). From es (esum originally for sum) came the present, the imperfect, and the first future tenses; from fuo came the perfect, the pluperfect, and second future tenses. The verb sum has neither gerund nor supine, and is in other respects defective, as appears from the paradigm just given. Sum takes before it certain prepositions, and is modified by them in its meaning; thus, with ad, adsum, it means I am at or near; with ab, absum, it means I am from, away from, absent; with pro, prosum, it means I am for, that is, I aid or benefit. In prosum, the letter d is inserted to prevent the hiatus which would be caused if two vowels came in succession; thus, pro-(d)-es, pronounced prodes; also prodest, proderam, prodero, prodessem. From the root mentioned above-namely, fu, fuo-come two forms not so common as those given in the table-namely, forem and fore; forem (-es, -et; -emus, -etis, -ent) is the imperfect subjunctive, and signifies I might be; corresponding to essem of the table; fore is the infinitive future, to be about to be; corresponding with the futurum esse of the table. VOCABULARY. Absens, part., being Hodie, to-day. absent. I am absent. Dum, conj., while. Intersum, interfui, in teresse (E. R. interest), I am among, I am concerned, I take an interest or part in. Ita, so. Longe, far. Magistratus, -ûs, m., a magistrate or go vernor. | Nisi, conj., unless. Oratio, onis, f., a speech (E. R. orator). Peregre, abroad. Præsum, præfui, præesse, I am before, I preside over, command Prosum, profui, prodesse, I am for, I am useful, I do good to. Pugna, -æ, f., a fight (E. R. pugilist). Quamdiu, as long as, how long? Quum (pronounced cum), conj., when, from the time when. Ubi, adv., where, when. Ut, as. Observe that these compounds of sum require their object to be in the dative case; as, prodest MIHI, he does good TO ME, or he benefits ME. EXERCISE 73.-LATIN-ENGLISH." 1. Deus omnibus locis adest. 2. Parvi pretii (of little avail), sunt arma foris, nisi est consilium domi. 3. Contemnuntur ii qui nec sibi nec alteri prosunt. 4. Ut magistratibus leges, ita populo præsunt magistrātus. 5. Ratio et oratio conciliat inter se homines, neque ulla re longius absumus a natură ferārum. 6. Ego sum lætus, tu es tristis. 7. Si sorte vestra contenti estis, beati estis. 8. Dum nos in schola erāmus, sorōres nostræ in horto erant. 9. Quum Carolus heri domi nostræ erat, ego peregre eram. 10. Quamdiu tu et frater tuus domi nostræ eratis, tu lætus eras, sed frater tuus erat tristis. 11. Quamdiu tu aberas, ego eram tristis. 12. Cur heri in schola non fuisti? 13. Quia cum patre peregre fui. 14. Quamdiu tu et pater tuus domo abfuistis? 15. Sex menses abfuimus. 16. Cur milites nostri pugnæ non interfuerunt? 17. Quia longius abfuerunt. 18. Ubi heri fueras quum domi tuæ eram ? EXERCISE 74.-ENGLISH-LATIN. 1. I am useful to thee. 2. Thou art useful to me. 3. The boys are not useful to (their) mothers. 4. Why are the girls not useful to (their) fathers? 5. When thou wast absent, I was sad. 6. How long has your father been absent ? 7. Charles took part in the fight. Wast thou yesterday at my house? 9. I shall be at thy house to-day. 10. Unless thou art happy at home, thou art not joyful abroad. 8. Pretium, -i, n., a re ward (E. R. price, precious). Propterea, on account of Provideo, 2, I see beforehand, foresee (E. R. provide). Prudens, -tis, prudent. Quod, because. Nisi, conj., unless, if Rectus, -a, -um, right. not. Obsum, obfui, obesse, I am in the way of, I oppose, I injure. Postea, afterwards. Præteritus, -a, -um, past. EXERCISE 75.-LATIN-ENGLISH. 1. Quamdiu felix eris, multi tibi erunt amici. 2. Pugna fuit atrocissima, propterea quod utriusque exercitus milites fortissimi fuērunt. 3. Ante belli initium in urbe fueramus. 4. Demosthenis ætate multi oratores magni et clari fuerunt, et antea fuerant, nec postea defuerunt. 5. Hæc res non profuit nobis sed obfuit. 6. Si quis virtutis compos erit, semper beatus erit. 7. Quamdiu sorte mea contentus ero, ero felix. 8. Actio recta non erit, nisi recta fuerit voluntas. 9. Si probi fuerimus, non deerit hominum laus. 10. Attenti este, discipuli. 11. Homines mortis memores sunto. 12. Contenti estote sorte vestră! 13. Mi fili, semper virtutis præceptorum memor esto! 14. Vir prudens non solum præsentia curat, sed etiam præterita mente repetit, et futura ex præteritis providet. 15. Boni bonis prodesse student. EXERCISE 76.-ENGLISH-LATIN. 1. Our soldiers were very brave in the fight. 2. Why were our soldiers braver than yours in the fight? 3. So long as you are happy, friends will not fail you. 4. Friends fail the wretched, 5. Before the beginning of the fight, I was in the city. 6. The brave will always be useful to the brave. 7. My enemies injure me. 8. If you are partakers of virtue, you will be happy. 9. So long as I am content with my lot, I shall be happy. 10. O scholars, you ought to be attentive in school! 11. They endeavour to be very brave. 13. 12. Be brave, my son. Prudent men foresee the future (pl.) from the past. VOCABULARY. Et-et, and-and (and Prius, adv., before. In, towards. norant. Non sum nescius, I am aware of. Quantum, how much. Qualis, -e, of what kind | Quantus, -a, -um, (E. R. quality). how great (E.R. quanQua mente sis, of tity). what disposition you are, what your feeling is. Scio, 4, I know (E. R. science). Tum, them. Observe that in indirect questions the dependent verb must be in the subjunctive (or dependent) mood; as, for example, narra mihi ubi fueris, tell me where you have been. Such a form is called an indirect question. The direct question would stand thus-ubi fuisti ? narra mihi, where hast thou been? tell me. In the latter case the question is direct, and the verb, as not being dependent, is in the indicative mood; but put narra mihi first, and then your question is implied rather than stated; it is, therefore, an indirect question. In both direct and indirect questions the English is in the indicative; consequently, in putting the dependent verb into English, you must in English use the indicative mood; but in putting the dependent verb into Latin, you must in Latin use the subjunctive mood. Compare what is said of the sequence of tenses, and similar and dissimilar tenses, in the last lesson. EXERCISE 77.-LATIN-ENGLISH. 1. Non sum nescius qua mente tu in nos sis. 2. Scio qua mente tu in nos semper fueris. 3. Non sum nescius qua mente tu et prius in nos fueris et nunc sis. 4. Non eram nescius qua mente tu in nos esses. 5. Scio quam sint incerti animi hominum. 6. Cogita quam brevis sit vita! 7. Qualis sit animus, ipse animus nescit. 8. Cogita quantum nobis bona exempla prosint. 9. Incertus sum ubi frater meus nunc sit. 10. Incertus sum ubi amicus meus et fuerit et nunc sit. 11. In certus eram ubi heri esses. 12. Narra nobis ubi heri fueritis. EXERCISE 78.-ENGLISH-LATIN. LESSONS IN GEOMETRY.—XX. THERE is a curious connection between the proportions of the sides of a hexagon and a heptagon inscribed in the same circle, and that is, that the length of the side of the heptagon is equal to the perpendicular let fall from the centre of the circle on any side of the hexagon. This may be seen from the following problem: PROBLEM LII.-To inscribe a heptagon in a given circle. Let A CE (Fig. 74) be the given circle in which it is required to describe a heptagon. Draw any diameter A K, passing through L, the centre of the given circle A CE; and from one of its extremities A as centre, with the distance A L, describe the are BL M, cutting the circumference of the circle ACE in the points в and м. Join B м, cutting A K at right angles in the point N. From the point B as centre, with the distance B N, draw arcs cutting the circumference of the circle A CE in c and H. Join C B, B H. These straight lines are sides of a heptagon inscribed in the circle A C E, and the heptagon itself may be completed by applying the compasses, with an opening equal to B N or N M, round the circumference of the circle A C E, as may be needful. The triangle A B L is an equilateral triangle (see Problem XVII., Vol. I., page 209), and A B is the side of a hexagon that may be inscribed in the circle A CE, in which the heptagon BCDEFGH has already been inscribed. K Fig. 74. H H A PROBLEM LIII.-To construct a heptagon on any given straight line. Let A B (Fig. 75) be the given straight line on which it is required to construct a heptagon. Produce A B indefinitely both ways to X and Y. Bisect A B in c, and again bisect C B in D. From B along the straight line B Y set off B E equal to five times B D, and from A along the straight line A F set off A X equal to в E, or five times B D. Then from the points A and B as centres, with the distances A E, B F respectively, describe the arcs E G, FG, cutting one another in the point &; and from a as centre, with a radius equal to A B, describe the arc HK, cutting the arcs FG, EG, in the points H and K. Join G H and G K, and bisect them respectively in the points, L, M; and join A M, B L, intersecting each other in the point N, which is the centre of the circle circumscribing the required heptagon. From the centre N, at the distance N A or N B, describe the circle A B KG H. Bisect the arcs A H, в K, in the points O, P, and join H O, O A, B P, P K. The figure A B PKG HO is a heptagon, and it is described on the given straight line A B, as required. X F N с D Fig. 75. PROBLEM LIV.-To construct an octagon on a given straight line. As it has been remarked in a former lesson (see page 192), it is easy to inscribe a hexagon in a given circle when we can place an equilateral triangle within it, as the process is merely to bisect the arcs intercepted between the ends of the sides of the triangle, and to form the hexagon by joining the six points thus obtained in the circumference of the circle. By a similar process of bisection, an octagon may be inscribed in a given circle when we have once placed a square within it; while the bisection of the arcs intercepted between the ends of the sides of a pentagon and hexagon will similarly produce a decagon and a dodecagon. There are, however, one or two processes of constructing an octagon on a given straight line which we give here, as they may be of use to the learner in cutting out an octagon in timber, or an octagonal flower-bed in turf. First, let A B (Fig. 76) be the given straight line on which it is required to construct an octagon. Produce A B indefinitely both ways to X and Y, and at the points A and B draw the straight lines A O, B P, perpendicular to A B or X Y. From A as centre, with the distance A B, describe the arc B C D, cutting A o in c, and X Y in D; and from B, as centre, with the distance B A, describe the arc A E F, cutting B P in E, and X Y in F. Join C D, E F, and bisect them in G and H respectively. Join A G, BH, and produce A G to meet the arc B C D in K, and в H to meet the arc A EF in L. Through the points K, L draw the Fig. 76. | straight lines K M, L N, parallel to A O or B P, and along K set off K Q, equal to ▲ B, and along L N set off L R, also equal to A B. Then from the points q and R as centres, with a radius equal to A B, draw arcs cutting the perpendiculars a 0, B P, in s and T. Join Q s, 8 T, T R. The figure A B LRT SQ K is an octagon, and it is described on the given straight line A B as required. Fig. 77. Next, let A B (Fig. 77) be the given straight line on which it is required to construct an octagon. Produce A B indefinitely both ways to X and Y, and through the points A and B draw the straight lines A O, B P, perpendicular to A B or X Y, and set off along A O and B P the straight lines A C, B D, each equal to A B. Join A D, B C, and produce them indefinitely to q and B respectively; and along D Q, CR, set off D E, C F, each equal to A B. Through E and F draw the straight lines E G, F H, meeting XY in G and H; and along E G, F H, set off E L and F K, each equal to A B. Through K draw Kи parallel to A Q, and cutting BP in M; and through L draw L N parallel to B R, and cutting AO in N. Join A K, B L, EM, MN, NF. The figure A B LEMN FK is an octagon, and it is described on the given straight line AB, as required. If it be required to cut off the corners of a square piece of wood or pasteboard as A B C D in Fig. 78, so as to form an octagon, first draw the diagonals a C, B D, intersecting each other at right angles in E, and then from the points A, B, C, D in succession, with the distances A E, BE, CE, DE, describe the arcs F E G, HE K, L E M, and N E O, having their terminations in the sides of the square. Then join o H, GL, K N, and O M F. The figure M F O H G L K N is an octagon. PROBLEM LV.-To inscribe a nonagon in a given circle. Let A B C (Fig. 79) be the given circle D M G Fig. 78. in which it is required to inscribe a nonagon. Draw any diameter, CE passing through the centre D of the circle A B C, and produce it indefinitely towards F. From the point E as centre, with the distance E D, describe the arc A D B, cutting the circumference of the circle A B C in the points A and B. Join A B, and pro duce it indefinitely both ways towards G and H, and let it cut CF at right angles in the point K. Then from X as centre, with a radius equal to DE, describe the semicircle LM N, having its terminations L, N, in the straight line GE; and from L and N as centres, with the radii L K, N K respec tively, describe the arcs KO, K P, meeting the semicircle L in the points o, P. Join D O, D P, cutting the circumference of the circle A B C in the points Q, E. Join A Q, QR, E B. These three straight lines are the sides of a nonagon inscribed in the circle A B C, which may be completed by following the same process with the arcs A C, C B, or, applying the compasses round the circumference of the circle with an opening equal to A Q, QR, or R B, and joining the points thus obtained. The straight line A B is manifestly the side of an equilateral triangle inscribed in the circle A B C, and the process which has been gone through is simply the trisection of the arc A B, or, what is virtually the same thing, the trisection of the angle A D B. F Fig. 79. |