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bers) the whole will be equal. 3. If from equal things we take equal things, the remainder will be equal, and the reverse in respect to unequal things. 4. The whole is greater than any of its parts. 5. Two right lines do not contain a space. 6. All the angles within a circle cannot amount to more nor less than 360 degrees, nor in a semicircle to more nor less than 180 degrees. 7. The value, or measure, of an angle is not affected or changed by the lines whereby it is formed being either lengthened or shortened. 8. Two lines standing at an angle of 90 degrees from each other will not be affected by any change of position of the entire figure in which they meet, but will still be mutually perpendicular.
After thus much preparation, we may conclude the student to be ready to proceed in the solution of problems, which we shall study to exhibit in the most simple, as well as in a progressive Inanner.
To imitate a given angle at a given point. Fig. 8. Let A B C be the given angle, and o the point on the line Q D whereon it is to be imitated. Draw the line A C, and from O measure towards D with an opening equal to A B: then from O make a seg. ment with an opening equal to B C, and from K make a segment with an openin equal to A C; their intersection at E off ive the point through which a line from § will make an angle with OD equal to the angle A B C.
To draw a parallel through a given point. Fig. 10. From the end, on any part of the given line A B, draw an oblique line to the given point C. Measure the angle made by A B C, and return another of equal measurement upon the line B C, so as to make the angle B C D equal to A B C : the line C D will be parallel to the line A. B. Or, as in fig. 11, you may from any points, say C. D, in the line A B, draw two semicircles of equal dimensions; the tangent E F will be parallel to A B. Or you may, according to Problem 5, draw a perPendicular from the given point to the given line, and draw another line through the given point at right angles with the perpendicular, proceeding from it to the line whose parallel was to be made, and which will be thus found. See fig. 12.
Parallelograms of equal base and altitude are reciprocally equal. Fig. 13. The parallelogram No. 1, is rectangular: No. 2 is inclined, so as to hang over a space equal to the length of its own base; but the line A. B., which is perpendicular thereto, divides it into two equal parts; let the left half, A B E, be cut off, and it will, by being drawn up to the right, be found to fit into the dotted space A C D. This theorem might be exemplified in various modes; but we presume the above will suffice to prove its validity.
of the parallelogram, and set it off on the line E. D. Draw F H parallel to E D, and H E parallel to D F, which will complete the parallelogram E F D H, equal to the triangle B A C. Prio Blexi xIV.
To apply a parallelogram to a given right line, equal to a given triangle, in a given right line figure. Fig. 17. Let A B be the given line to which the parallelogram is to be annexed. Let C be the triangle to be commuted, and D the given angle. Make B E F G equal to C, on the angle E B G : continue A B to E ; carry on FE to K, and make its parallel H A L, bounded by F H, parallel to E A: draw the diagonal H K, and G M both through the point B; then K L, and the parallelogram B M A L will be equal to the triangle C, and be situated as desired.
To make a parallelogram, on a given inclination, equal to a right-lined figure. Fig. 18. Let A B C D be the right-lined figure, and F K H the given angle or inclination; draw the line D B, and take its length for the altitude, F K, of the intended parallelogram, applying it to the intended base line K M : now take half the greatest diameter of the triangle D C B, and set it off from K to M, and set off half the greatest diameter of the triangle D A B, and set it off from H to M ; make G H to LM parallel to FK, and FG parallel to K H. The parallelogram F K G H will be equal in area to the figure A B C D, and stand at the given inclination or angle.
PRO is Lexi XVI.
To describe a square on a given line. Fig. 19. Raise a perpendicular at each end of the line A B equal to its length ; draw the line CD, and the square is completed.
- The OREM xvii.
The square of the hypothenuse is equal to both the squares made on the other sides of a right-angled triangle. Fig. 20. This comprehends a number of the foregoing propositions, at the same time giving a very beautiful illustration of many. Let A B C be the given right-angled triangle ; on each side thereof make a square. For the sake of arithmetical proof, we have assumed three measurements for them : viz. the bypothenuse at 5, one other side at 4, and the last at 3. Now the square of 5 is 25. The square of 4 is 16, and the square of 3 is 9: it is evident the sum of the two last sides make up the sum of the hypothenuse's square; for 9 added to 16 make 25. But the mathematical solution is equally simple and certain. The squares are lettered as follow : B D C E, F G B A, and A H G K. Draw the following lines; FC, B K, A D, A L, and A E. We have already shown, that parallelograms and triangles of equal base and altitude are respectively equal. The two sides F B, B C, are equal to the two sides A B, BD, and the angie D A B is equal F B C : the triangle A B D must therefore be equal to the angle FB C. But the parallelogram B L is double the triangle A B D. The square G B is also double the triangle F B C : consequently the parallelogram B L is equal to the square G B. The square H C in like manner is proved to be equal to the parallelogram C. L., which completes the solution. Euclid, 47th of 1st Book.
To draw a spiral line from a given point. Fig. 37. Draw the line AB through the given point C, and from C draw the semicircle DE, and then shift to D for a centre, and make the semicircle AE in the opposite side of the line: shift again from D to C for a centre, and draw the semicircle FG ; and then continue to change the centres alternately, for any number of folds you may require ; the centre C serving for all above, the centre D for all below, the line AB. With respect to the application of geometry to its pristine intent, namely, the measurement of land, we must refer our read, rs to Svavey ING : under which head it will be found practically exemplified. We trust sufficient has been here said to show the utility and purposes of this important science, and to prove serviceable to such persons as may not have occasion for deep research, or for extensive detail. GEORGIC, a poetical composition upon the subject of husbandry, containing rules therein, put into a pleasing dress, and set off with all the beauties and embellishments of poetry. GEORGINA, in botany, a genus of the Syngenesia Superflua class and order. Ireceptacle chafty ; no down; calyx double; the outer many-leaved; inner oneleaved, eight-parted. There are three species. GERANIUM, in botany, crane’s bill, a genus of the Monadelphia Decandria class and order. Natural order of Gruinales. Gerania, Jussieu. Essential character: calyx five-leaved; corolla five-petalled, regular; nectary five honied glands, fastened to the base of the longer filaments; fruit five-grained, beaked; beaks simple, naked, neither spiral nor bearded. There are thirty-two species. There are five species indigenous to the United States. The root of one of these, G. maculatum, or spotted crane's bill, is an astringent, and the decoction of it, made with milk, is useful in cholera infantum. GERARDIA, in botany, so called in honour of John Gerarde, our old English botanist, a genus of the Didynamia Angiespermia class and order. Natural order
of Personate. Scrophularize, Jussieu. Essential character: calyx five-cleft; corolla two-lipped, lower lip three-parted, the lobes emarginate, the middle segments two-parted; capsule two-celled, gaping. There are ten species. GERMINATION. When a seed is placed in a situation favourable to vegetation, it very soon changes its appearance; the radicle is converted into a root, and sinks into the earth; the plumula rises above the earth, and becomes the trunk or stem. When these changes take place, the seed is said to germinate; the process itself has been called germination, which does not depend upon the seed alone; something external must affect it. Seeds do not germinate equally and indifferently in all places and seasons; they require moisture and a certain degree of heat, and every species of plant seems to have a degree of beat peculiar to itself, at which its seeds begin to germinate; air also is necessary to the germination of seeds; it is for want of air, that seeds which are buried at a very great depth in the earth either thrive but indifferently, or do not rise at all. They frequently preserve, however, their germinating virtues for many years within the bowels of the earth; and it is not unusual, upon a piece of ground being newly dug to a considerable depth, to observe it soon after covered with several plants, which had not been seen there in the memory of man. Were this precaution frequently repeated, it would perhaps be the means of recovering certain species of plants which are regarded as lost; or which, perhaps, never coming to the knowledge of botanists, might hence appear the result of a new creation. Light is supposed to be injurious to the process, which affords a reason for covering the seeds with the soil in which they are to grow, and for carrying on the business of malting in darkened apartments, malting being nothing more than germination, conducted with a particular view. GEROPOGON, in botany, a genus of the Syngenesia Polygamia Aequalis class and order. Natural order of Compositz Semiflosculosae, or compound flowers, with semi-florets or ligulate florets only. Cichoraceae, Jussieu. Essential character: calyx simple; receptacle with bristle-shaped chaffs; seeds of the disk with a feathered down of the ray, with five awns. There are three species. GESNERIA, in botany, so named in honour of Conrad Gesner, of Zurich, the famous botanist and natural historian, a genus of the Didynamia Angiospermia,