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pearances observed in these rocks it is concluded, that the waters in which they were formed had risen with great rapidity, and had afterwards settled into a state of considerable calmness.

The collections and deposits derived from the materials of pre-existing masses, worn down by the powerful agency of air and water, and afterwards deposited on the land or on the sea coasts, are termed alluvial, and are, of cours, of much later formation than any of the preceding classes. These deposits may be divided into, 1. Those which are formed in mountainous countries, and are found in vallies, being composed of rolled masses, gravel, sand, and sometimes loam, fragments of ores, and different kinds of precious stones. 2 Those which occur in low and flat countries, being peat, sand, loam, bog iron oar, nagelflech, calc-tuff, and calcsinter: the three latter being better known by the names breccia, tufa, and stalactite.

In this ingenious system, in which so much knowledge of the subject prevails, and in which the marks of long and patient investigation are evident, a very close accordance with geological facts is generally observable. Some few difficultics however occur, particularly it seems with respect to the new trap formation; since, although the appearances which this is intended to explain do not better agree with any other supposition, still the rising of the waters, whilst they yet covered the summits of primitive mountains, has much the appearance of a supposition made up for this particular purpose; and as, at the same time, it appears to be warranted by no other phenomena, it seems to require some further consideration, before it is fully admitted.

For more particular observations on the various characters, and on the different classes of rocks, see Rocks.

GEOMETRA, in natural history, one of the families of the Phalana genus of insects. See PHALENA.

GEOMETRY, in its original sense, related simply to the measurement of the earth, and was invented by the Egypti. ans, whose lands being annually inundated, required to be frequently measured out to the respective owners, so that each might repossess his property. seems probable, that in the operations attendant on that act of justice, many discoveries were made relating to the properties of figures, which gradually led on to an extension of the science, and to

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the cultivation of the arts of navigation and astronomy, which, indeed, first flou rished in that quarter. We are rather in the dark as to many improvements made in the infancy of geometry, and its at tendant speculations; many tracts of supposed value having been entirely lost, though some faint traces and fragments of their subjects, if not of their contents, have from time to time been discovered. The Grecians appear to have been enthu siasts in their reception of the new science; accordingly we find that Thales, Pythagoras, Archimedes, Euclid, &c. exerted themselves to instruct their countrymen, and thus to prepare the way for the philosophy of Ptolemy, Copernicus, and others of the ancient school; and of Des Cartes, Leibnitz, and the immortal Newton, in our more enlightened times. At present, geometry is justly considered to be the basis of many liberal sciences, and to be an indispensable part of the education of those who purpose exer cising even the more mechanical arts to advantage.

We shall submit to our readers a general view of this most useful and fascinating attainment, and, by a gradual display of its rudiments, open the field to further advancement, which may be easily insured, by consulting those authors who have become eminent for the display of whatever relates to the superior branches of geometry. In the first instance, we shall submit the following definitions, as laid down by Euclid in his Elements, recommending them to the serious attention of the student; they being absolutely ne. cessary towards his competent appreciation and understanding of the succeeding propositions.

DEFINITIONS.

1. A point hath neither parts nor mag. nitude. 2. A line has length, without breadth. 3. The ends, or bounds, of a line are points. 4. A right line lies evenly between two points. 5. A superficies or plane has only length and breadth. 6. Planes are bounded by lines. 7. A plain superficies lies evenly and level between its lines. 8. A plain angle is formed by the meeting of two right lines. 9. When an angle measures 90 degrees, it is called a right angle. 10. When less than 90 degrees, it is said to be an acute angle. 11. When more than 90 degrees, it is called an obtuse angle. 12. A term, or bound, implies the extreme of any

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thing. 13. A figure is contained under one or more bounds. 14. A circle is a plain figure, contained in one line, called the circumference, every where equally distant from a certain point within it. 15. That equi-distant point within the circle is called its centre. 16. A line passing from one side to the other of a circle, and through its centre, is the greatest line it can contain, and is called its diameter. 17. The diameter divides the circle into two equal and similar parts, called semi-circles. 18. When a line shorter than the diameter is drawn from one point to another on the circumference of a circle, it is called a chord. 19. The part of the circle, so cut off or divided by such line or chord, is called an arc or segment. 20. Figures contained under right lines are called rightlined figures. 21. A figure having three sides is called a triangle. 22. If all the sides of a triangle are of the same length, it is called an equilateral triangle. 23. If all the sides and angles are unequal, it is called a scalene triangle. 24. If two of the sides are of equal length, it is called an isosceles, or equi-crural triangle. 25. If containing a right angle, it is called a right-angled triangle. 26. The long side subtending, and opposite to, the right angle, is called the hypothenuse. 27. When the two shortest sides of a triangle stand at a greater angle than 90 degrees, the figure is said to be "obtuse;" and when all the angles are acute, it is called an acute angled triangle. 28. When two lines preserve an equal distance from each other in every part, they are said to be parallel. 29. Parallel lines may be either straight or curved, but can never meet. 30. A figure having four equal sides, and all the angles equal, is a square. 31. But if its opposite angles only be equal respectively, the figure will then be a rhombus, or lozenge. 32. When all the sides of a figure are right lines, and that the opposite sides are parallel and equal, it is called a parallelogram. 33. If the opposite sides are equal, the others being unequal, the figure is called a rhomboides. 34. Foursided figures unequal in all respects, are called trapesia. 35. Figures having more than four sides are called polygons, and are thus distinguished: with five sides, it is called a pentagon; with six, an hexagon; with seven, an heptagon; with eight, an octagon; with nine, an enneagon; with ten, a decagon; with eleven, an endecagon; with twelve, a dodecaVOL, V.

gon. 36. A solid has length, breadth, and thickness. 37. A pyramid is a solid standing on a base, of any number of sides, all of which converge from the base to the same point or summit. 38. When standing on a triangular base, it is called a triangular pyramid; on four, a square pyramid; on five, a pentagonal; and thus in conformity with the figure of its base. 39. Every side of a pyramid is a triangle. 40. A cone is found by the revolution of a triangle on its apex, or summit, and a point situated in the centre of its base; therefore a cone (like a sugar-loaf) has a base, but no sides. 41. A prism is a figure contained under planes, whereof the two opposite are equal, similar, and parallel; and all the sides parallelograms. 42. A sphere is a solid figure, generated by the revolution of a circle on its diameter, which is then called the axis. 43. A cube is a solid formed of six equal and mutually parallel sides, all of which are squares. 44. A tetrahedron is a solid contained under four equal, equilateral triangles. 45. A dodecahedron is a solid contained under twelve equal, equilateral, and equiangular pentagons. 46. An icosahedron is a solid contained under twenty equal, equilateral triangles. 47. A parallelopipedon is a figure considered under six quadrilateral figures or planes, whereof those opposite are respectively parallel. 48. Figures, or bodies, are said to be equal, when their bulks are the same; and similar, when they are alike in form, though not equal. 49. Therefore similar figures or bodies are to each other in proportion to their respective areas or bulks. 50. The line or space on which a figure stands is called its base; its altitude is determined by a line drawn parallel to its base, and touching its vertex, or highest part. lined figure is said to be inscribed within another, when all its projecting angles are touched thereby. 52 The figure surrounding or enveloping another is said to be described around, or on it. 53. When a line touches a circle, and proceeds without cutting it, such line is called a tangent. 54. Any portion less than a semicircle, taken out from a circle by two lines, or radii, proceeding from the centre, is called a sector.

51. A right

Certain AXIOMs are likewise proper to be carried in mind; viz. 1. That things equal to one and the same thing are equal to one another. 2. If to equal things (or numbers) we add equal things, (or num

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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

manner.

PROBLEM I.

To describe an equilateral triangle upon a given line. Let A B (fig. 1.) be the given line, with an opening of your compasses equal to its length: from each end, A and B, draw the arcs C D and E F, to whose point of intersection at C draw the lines AC and B C.

PROBLEM II.

To divide an angle equally. Fig. 2. Let BAC be the given angle, measure off equal distances from A to B, and from A to C; then with the opening B C draw alternately from B and from C the arcs which intersect at D: a line drawn from A to D will bisect the angle BA C.

PROBLEM 111.

To bisect a given line. Fig. 3. Let A B be the given line; from each end (or nearer, if space be wanting,) with an opening of your compasses rather more than half the length of A B, describe the arcs which intersect above at C, and below at D: draw the line C D, passing through the points of intersection, and the line A B will be divided into two equal parts. Observe, this is an easy mode of erecting a perpendicular upon any given line.

PROBLEM IV.

To raise a perpendicular on a given point in a line. Fig. 4. With a moderate open. ing of your compasses, and placing one of its legs a little above or below the given line, describe a circle passing through the given point A on the line B C ; then draw a line from the place where the cir cle cuts at D, so as to pass through E, the centre to F on the opposite side of the circle: the line F A will be the perpendi cular required.

PROBLEM V.

From a given point to let fall a perpendi cular on a given line. Fig. 5. From the given point A draw the segment B C passing under the line DE; bisect B C in F, and draw the perpendicular A F.

THEOREM VI.

The opposite angles made by intersecting lines are equal; (fig. 6.) as is shown in this figure: 0, 0, are equal; p, p, are equal; 9, s, are equal.

PROBLEM VII.

To describe a triangle with three given lines. Fig. 7. Let A B, B C, and C D, be the three giyen lines; assume either of them, say A B, for a base; then with an opening equal to B C, draw the seg ment from the point B of the base, and with the opening C D make a segment from C: the intersection of the two segments will determine the lengths of the two lines B C and C D, and of the angle A B C.

PROBLEM VIII.

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 O 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 segment with an opening equal to B C, and from K make a segment with an opening equal to A C; their intersection at E will give the point through which a line from Ŏ will make an angle with O D equal to the angle A B C.

THEOREM IX.

All right lines severally parallel to any given line are mutually parallel, as shown in

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fig. 9, where A B, C D, E F, and G H, being all parallel to I K, are all parallels to each other severally.

N. B. They all make equal angles with the oblique line O P.

PROBLEM X.

To draw a parallel through a given point. Fig. 10. From the end, on any part of the given line AB, 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 CD, 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.

THEOREM XI.

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.

THEOREM ΧΙΙ

Triangles of equal base and altitude are reciprocally equal. Fig. 14. As every parallelogram is divisible into two equal and similar triangles, it follows that the same rule answers for both those figures under the position assumed in this proposition: we have shown this by fig. 15.

PROBLEM XIIT.

To make a parallelogram equal to a given triangle, with a given inclination or angle. Fig. 16. Let B A C be the given triangle, and E D F the given angle. On the line DF measure a base equal to B C, the base of the triangle. Take B G equal to half the altitude of the triangle for the altitude

of the parallelogram, and set it off on the line ED. Draw F H parallel to E D, and HE parallel to D F, which will complete the parallelogram E F D H, equal to the triangle B AC.

PROBLEM 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 BEFG equal to C, on the angle E B G; continue A B to E; carry on F E to K, and make its parallel H A L, bounded by FH, parallel to E A: draw the diagonal HK, and G M both through the point B; then K L, and the parallelogram B MAL will be equal to the triangle C, and be situated as desired.

PROBLEM XV.

To make a parallelogram, on a given in. clination, 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 DB, 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 DC 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 L M parallel to F K, 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.

PROBLEM 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 C D, and the square is com pleted.

THEOREM 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 hypothe. nuse's square; for 9 added to 16 make 25. But the mathematical solution is equally simple and certain. The squares are lettered as follow: BDCE, FG BA, and AHGK. 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 angle D A B is equal FBC: the triangle ABD must therefore be equal to the angle FB C. But the parallelogram BL is double the triangle ABD. The square G B is also double the triangle FBC: consequently the parallelogram BL 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.

PROBLEM XVIII.

To divide a line, so that the rectangle contained under the whole line and one segment be equal to the square of the other segment. Fig. 21. On the given line A B describe the square A B C D; bisect A Cin E, and with the distance E B extend A C to F, measuring from E. Make on the excess FA the square F H, and continue G H to K. The square F H will be equal to the parallelogram H D.

PROBLEM XIX.

To make a square equal to a given rightlined figure. Fig. 22 Let A be the given right-lined figure: commute it to a paral lelogram, B D, as already shown, (prob. 15) add the lesser side E D to BE, so as to proceed to F: bisect B F in G, and from that point describe the semicircle BHF. Continue D E to H, which will give H E for the side of a square equal in area to the parallelogram B D, and to the original given figure A.

PROBLEM XX.

To find the centre of a given circle. Fig. 23. Draw at pleasure the chord A B, bisect it in D by means of a diameter, which being bisected will give F for the centre of the circle.

PROBLEM ΧΧΙ.

Fig. 24. Let A B C be the given segment: draw the line A C, and bisect it in D; draw also the perpendicular B E through D, draw B A, and on it make the angle BA E, equal to DB A; this will give the point of intersection E for the centre, whence the circle may be completed. It matters not whether the segment be more or less than a semicircle.

PROBLEM XXII.

To cut a given circumference into two Draw the line AB, equal parts. Fig. 25. bisect in C, the perpendicular DC will divide the figure into two equal and similar parts.

PROBLEM XXIII.

In a given circle to describe a triangle equiangular to a given triangle. Fig. 26. Let A B C be the circle, and D E F the triangle given. Draw the line G H, touching the circle in A: make the angle HAC equal to D E F, and G A B equal to DFE: draw B C, and the triangle B A C will be similar to the triangle D E F.

PROBLEM XXIV.

About a given circle to describe a triangle similar to a given triangle. Fig. 27. Let ABC be the given circle, and D E F the given triangle: continue the line E F both ways to G and H, and having found the centre K, of the circle, draw a radius, K B, at pleasure; then from K make the angle BK A equal to D E C, and BK C equal to D F H; the tangents L N perpendicular to K C, M N perpendicular to KB, and M L perpendicular to K A, will form the required triangle.

PROBLEM XXV.

To describe a circle about a given triangle. Fig. 28. In the given triangle ABC, bisect any two of the angles; the intersection of their dividing lines, B D and C D, will give the centre D, whence a circle may be described about the triangle, with the radius DC.

PROBLEM XXVI.

To inscribe a circle in a given triangle. Fig. 29. In the triangle A B C, divide the angles A B C, and B C A, equally by the lines B D, C D. Their junction at D will give a point whence the circle E C F may be described, with the radius D F per

To complete a circle upon a given segment. pendicular to B C.

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