2. Draw a line EF equal to one of the lines as AB, produce EF to G, so that F G shall be equal to CD, the other line. 3. Bisect EG in the point 0, from which as a centre, with OE as a radius, describe the arc EHG. 4. Erect a perpendicular to EG from the point F, till it meets the arc of the semicircle in H; when FH shall be the mean proportional required; for A B shall have the same ratio to F H as FH has to CD. EXAMPLES. 1. Find a mean proportional between two lines 4 inches and 1 inch long. The result will be a line 2 inches long. 2. Required a line that shall have the same ratio to a line 3 inches long, as another line 14 inches long bears to the line required. PROBLEM XX. To divide a line into extreme and mean proportion. 1. Let A B be the given line. 2. From one end of the line as B, erect a perpendicular B C, equal to half A B. Draw the hypothenuse CA. 3. From C as a centre, with CB as a radius, describe the arc B D cutting CA in D. E 4. From A as a centre, with AD as a radius, describe an arc cutting A B in E; when A B shall be divided in the ratio required; for the lesser segment EB shall have the same proportion to the greater segment A E, as AE has to the sum of the two segments, A B. EXAMPLE. Divide a line 4 inches long into extreme and mean ratio. PROBLEM XXI. To find the centre of a given circle. 1. Describe any circle, and conceive the centre to be imperceptible to the naked eye. 2. Draw a right line or chord in any direction; say as EC, in Prob. XI. Bisect EC by a perpendicular, as in Prob. IX. This perpendicular (produced if necessary) will furnish another chord, which will be the diameter of the circle. 3. Bisect the diameter by Prob. IX. when the point of bisection will be the centre of the circle. PROBLEM XXII. To draw a tangent to a circle from a point without (i. e. outside) the circumference. 1. Describe a circle and fix upon the point A as the given point. 2. Draw a line from A to O, the centre of the circle. Upon A O as a diameter describe a semicircle, cutting the circle in the point D. 3. Draw AD C, and it shall be the tangent required, of which D is the point of contact. PROBLEM XXIII. To draw a tangent to a circle that shall pass through a point in the circumference. 1. Describe a circle as before, and let D be the given point. 2. Draw a line from D to the centre of the circle O. 3. Make A D C at right angles to OD, and it shall be the tangent required. PROBLEM XXIV. To draw a tangent to any point in the arc of a circle when the centre is inaccessible. 1. Let A CB be the given arc, and A the given point. 2. Draw a chord from A to B. Bisect it in the point D; from which make D C perpendicular to A B, cutting the arc in C. Draw the chord C A. 3. Draw the line TAO, making the angle CAT equal to the angle CAD; when TAO shall be the tangent required. A rectilineal tangent to a circle is always at right angles to that radius which determines the point of contact, i. e. TAO is at right angles to A E, the point E being the centre of the arc A C B, and A the point of contact. PROBLEM XXV. Any right line being given, and a point without another line at right angles to the extremity of the first line also given; to determine the rise of the arc of a circle which shall touch the said extremity and the said point. 1. Let A be the extremity of the line O A, and B the given point without the line A E, as in Prob. XXIV. 2. Draw the line A B; bisect it in D; draw D C any length at right angles to A B. 3. Produce the line O A to T; and bisect the angle TAD by the line A C cutting D C in C, when D C shall be the rise required. In the same manner other points besides C may be determined to show the true curve of the whole arc AC B. For let AC be bisected by a perpendicular; bisect the angle TAC by a line drawn from A cutting the perpendicular, and this point of intersection will give the rise between A and C. Proceed in this way, bisecting as many chords and angles as may be necessary. Apply their results from C to B, after which the arc may be drawn by hand through all these points. It will be obvious that the two preceding problems may be rendered of considerable service in the projection of railway lines, when parts are to be curved. Tangents are of great importance both in geometry and philosophy. To draw them accurately, their curves should generally be done first, while "inking in," that the straight lines may be drawn from the extremities of the curves, instead of drawing the curves to meet the straight lines. PROBLEM XXVI. To make a circle that shall pass through three points, provided they are not in the same straight line. 1. Let A, B, and C, be the given points. 2. Join the points by the lines A B and B C. 3. Bisect A B and B C by lines (at right angles to each) cutting each other in the point O. 4 Take O as a centre and O A as a radius; describe a circle, and it shall pass through the given points as required. By this problem the ribs of a balloon may be made their proper shape, so that when joined, the requisite number may form a spherical figure. Also by it the diameter of a fly wheel may be determined, though only a small portion of its circumference may be accessible. PROBLEM XXVII. To describe a circle about a given triangle. 1. This problem being in effect the same as the preceding, consider the points A and C as joined by a straight line, when ABC will be the given triangle. 2. Bisect either of the two sides, as A B and B C by lines cutting each other in O, which will be the centre of the circle required. In every right-angled triangle, the centre O will fall on the hypothenuse, and will bisect it; in an acute-angled triangle it will fall within the triangle; and in an obtuse-angled triangle it will fall without the triangle. EXAMPLES. 1. Draw any acute-angled triangle, and describe a circle about it. 2. Make a triangle, its sides being 4 inches, 3 inches, and 1 inches, and circumscribe it by a circle. PROBLEM XXVIII. To draw a circular arch of a given span and rise. 1. Draw two lines for the given span and rise. 2. Make A B equal to the span. Bisect it in C, by the perpendicular CK, making C Kequal to the rise. Bisect 3. Draw the lines A K, K B. each by perpendiculars which shall cut each other in the point O. 4. From O as a centre, with OA as a radius, describe the arc AK B. 5. Produce OA and OB till AF and BI are equal to the depth of the arch From O with OF, stones or voussures. describe the arc F I. of equal parts, as 7. direction of the centre F B Divide it by trials into any odd number The centre stone K, is called by architects the Key stone of the arch. EXAMPLE. Draw a circular arch, (having 9 voussures of half an inch deep,) its span 3 inches, and its rise 1 inch. PROBLEM XXIX. To inscribe a circle in a triangle. 1. Let A C B be the triangle given. 2. Bisect any two angles, as CAB, CBA, by the lines AD, BD, cutting each other in the point D. 3. Let fall the perpendicular D E from D to AB. From D as a centre, with DE as a radius, describe a circle, which shall be inscribed in the triangle, as required. EXAMPLE. Inscribe a circle in any obtuse angled triangle, and examine its tangents. PROBLEM XXX. To inscribe a square in a circle. 1. Describe any circle, through which draw a diameter in any direction. Bisect this diameter by drawing another diameter at right angles to it; the circumference will then be cut into four equal parts. |