Edge (geometry)

“ We develop the surface into the plane of the base by turning each lateral face about its lower edge into the horizontal plane by the method used in § 14. If one face has been turned down, say ABV to ABP, then the point Q to which the vertex of the next face BCV comes can be got more simply by finding on the line V1Q perpendicular to BC the point Q such that BQ = BP, for these lines represent the same edge BV of the pyramid. ”

“ If the angle Red were smaller, and the angle Green larger, would the adjacent angles still be equal to two right angles?
Then,—
Any two adjacent angles are equal to two right angles.
If we draw the straight line c d where the edges of the square corners come together, what kind of angles will a c d, d c b, be? ”

“ Section 50 The area of a square is well known, as it is the multiplication of the vertical edge by the horizontal edge.
Section 51 In an ‘even footed’ (isosceles) triangle, the angles opposite the equal edges are themselves equal. For example, in triangle ABC, side AB is equal to AC, and angle ABE is equal to ACE. The proof to this is that if isosceles triangle BCD is constructed on the opposite side of line BC, and line AD is shared as a side of both triangles ABD and ACD, angles BAD and CAD will be equal to each other, as demonstrated in section 23. ”

“ The straightedge and the compasses are the only two drawing instruments recognized in elementary geometry. [51] While this involves more than Euclid's postulate, it is a better working assumption for beginners.
3. A straight line is the shortest path between two points. This is easily proved by the method of Euclid [52] for the case where the paths are broken lines, but it is needed as a postulate for the case of curve paths. It is a better statement than the common one that a straight line is the shortest distance between two points ”

“ Section 3 The lines on the sides of a plane are called edges, and the point at which they meet is called a vertex (lit. corner) .
Section 4 A plane is surrounded by not less than 3 edges. The type of shape of a plane is determined by its edges; if it has three edges it is called a triangle, if four a square, if five a pentagon and so on.
Section 5 According to the number of its edges is the number of its corners with the exception of a circle, which is enclosed with one line and has no corners. ”

“ Teachers will naturally see that just as we may measure the plane angle by taking the ratio of an arc to the whole circle, and of a dihedral angle by taking the ratio of that part of the cylindric surface that is cut out by the planes to the whole surface, so we may measure a polyhedral angle by taking the ratio of the spherical polygon to the whole spherical surface. ”

“ Similarly, if you draw the circle FGI from the center C, the amount of degrees in arc FG from circle FGI is the same as the degrees in vertex ABC according to arc DE from circle DEH. In the same manner, whatever the distance of the arc that lies between the two lines, it remains in proportion to the circumference, and the angle is the same, regardless whether the length of the lines are in cubits, handbreaths, or finger measurements.
Section 10 A right angle is one that contains 90 degrees and whose edges are perpendicular, not leaning towards either side. ”

“ We may, however, ignore the accented letters in translating theorems, for a configuration of lines and its polar with respect to a linear complex have the same projective properties. In Lie’s transformation a linear complex corresponds to the totality of spheres cutting a given sphere at a given angle. A most remarkable result is that lines of curvature in the sphere geometry become asymptotic lines in the line geometry. ”

“ For the exterior side, draw a line 360 yards in length, at the ends of which, lines are to be directed to the centre of the polygon, at the angle required; (vide Practical Geometry—To find the angles at the centre, and circumference of a regular polygon,) then bisect the exterior side, and draw the perpendicular, &c., &c., as described for the construction on a hexagon. ”

“ An angle at the circumference is measured by half the arc on which it stands.
Let B A D be an angle whose vertex is in the circumference of the circle whose centre is C; then will it be measured by half the arc B D.
For through the centre draw the diameter A E, and join the points C and B.
The exterior angle E C B is equal to the sum of the angles B and B A C.
Because the sides C A, C B, are radii of the circle, they are equal, the triangle is isosceles, the angles B and B A C opposite the equal sides are equal, and the angle B A C is half of both. ”

“ They are used for the purpose of testing and correcting any surfaces which are required to be made true. A straight edge is also used for testing the truth of surfaces; it is generally so called when its surface is very narrow as compared with its length, being usually the side face of a long flat bar. ”

“ If a right line making an angle with a right line given, and knit unto it with a base, be cut according to any rate assigned, a parallell to the base from the ends of the segments, shall cut the line given according to the rate assigned. ”

“ Section 15 All corners that can exist on one side of a line, as in this figure,
always add up to 180 degrees.
Section 16 The corner with the greatest angle has opposite it the longest side, and the reverse is true as well.
Section 17 The complement of an angle is the amount needed for an acute angle to reach 90 degrees.
Section 18 The remainder [שארית] of an angle is amount needed for an angle, whether acute or obtuse, to reach 180 degrees.
Section 19 Parallel lines are two lines that run next to each other separated by a constant distance. These lines will never intersect with one another. ”

“ Now the values of the sine, cosine, and tangent (functions of the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining them [Pg 235] by actual measurement, using the protractor and squared paper. Some of those needed for such angles as a pupil in geometry is likely to use are as follows:
It will of course be understood that the values are correct only to the nearest thousandth. ”

“ Any angle is easily trisected by means of certain higher curves, such as the conchoid of Nicomedes (ca. 180 B.C.) , the quadratrix of Hippias of Elis (ca. 420 B.C.) , or the spiral of Archimedes (ca. 250 B.C.) . But since this problem, stated algebraically, requires the solution of a cubic equation, and this involves, geometrically, finding three points, we cannot solve the problem by means of straight lines and circles alone. In other words, the trisection of any angle, by the use of the straightedge and compasses alone, is impossible. Special angles may however be trisected. ”

“ Now, the most elementary geometry teaches us that every triangle having these angles will have the lengths of its sides in the same invariable proportions one to another whatever may be the size of the triangle itself, and it has been found convenient to calculate these proportions once for all, not merely for angle 20°, but for every angle up to 90°. ”

“ A specimen of the form of template for drawing these curves is shown in Figure 213; g g, is the centre line parallel to the edges R, S; lines m, n, represent the diameter of the thread at the top, and o, p, that at the bottom or root; edge a is formed to the points (found by the constructions in the figures as already explained) for the tops of the thread, and edge f is so formed for the curve at the thread bottoms. The edge, as S or R, is laid against the square-blade to steady it while drawing in the curves. ”

“ They need to be defined for young people, but most of them are not basal, that is, they are not used in the proofs of propositions. Some of these terms, such as magnitudes, curve line, broken line, curvilinear figure, bisector, adjacent angles, reflex angles, oblique angles and lines, and vertical angles, need merely a word of explanation so that they may be used intelligently. ”

“ In plane geometry it is easily proved that if two circles intersect, the tangents from any point on their common chord produced are equal. For if the common chord is AB and the point P is taken on AB produced, then the square on any tangent from P is equal to PB × PA. The line PBA is sometimes called the radical axis.
Similarly in this proposition concerning spheres, if from any point in the plane of the circle formed by the intersection of the two spherical surfaces lines are drawn tangent to either sphere, these tangents are equal. ”

“ If a right line whose power is sesquialter unto the side of an equilater triangle, be cut after a double reason, the double segment perpendicular to the center of the triangle, knit together with the angles thereof shall comprehend a tetraedrum. 13 p xiij.
For a solid to be comprehended of right lines understand plaines comprehended of right lines, as in other places following. ”

“ Each face is a regular polygon, hence the angles of the faces at any vertex must be angles in equal regular polygons, must be together less than four right angles (XI. 21) , and must be three or more in number. Each angle in a regular triangle equals two-thirds of one right angle. Hence it is possible to form a solid angle with three, four or five regular triangles or faces. These give the solid angles of the tetrahedron, the octahedron and the icosahedron. The angle in a square (the regular quadrilateral) equals one right angle. ”

“ Place the point of angle on the centre of the semi-circle, or the centre of the semicircle on the point of the angle, and then by seeing how many degrees the opening of the angle measures on the graduated edge of the semicircle, you will find the size of the angle. If the sides of the angle do not extend to the circumference, you may extend them till they do. If they extend beyond the circumference, measure the angle where the graduated circle cuts the sides. ”

“ Connect the ends of the two lines by a straight line.
What kind of an angle is that opposite the last line drawn?
Are any two of its sides equal?
Write its two names under it.
Draw a horizontal straight line of any length.
At one end draw a vertical line of equal length.
Complete the triangle, and write two names inside.
Draw a right-angled triangle whose base is of any length, and its perpendicular twice as long.
Draw a right-angled triangle whose base is three times as long as any short measure, and its perpendicular five times as long as the same measure or unit.
54
Diagram 20. ”

“ This division is taken out of the 4 d j. of Euclide, where rectitude or straightnes is attributed to a line, as if from it both surfaces and bodies were to have it. And even so the rectitude of a solid figure, here-after shall be understood by a right line perpendicular from the toppe unto the center of the base. Wherefore rectitude is propper unto a line: And therefore also obliquity or crookednesse, from whence a surface is judged to be right or oblique, and a body right or oblique.
5. A right line is that which lyeth equally betweene his owne bounds: A crooked line lieth contrariwise. 4. ”

“ By joining V ́ A ́ and drawing T ́ S ́ parallel to V ́ F ́, then T ́ S ́ is obtained which will be the length of half the side of the square for drawing the inner spiral. The method for obtaining the inner spiral is the same as for the outer.
Fig. 37 There is no geometric means of drawing a perfect catenary curve; at best we can only obtain{196} it by an approximation in geometry. The curve is formed by suspending a chain from two points and pricking points along the curve of the chain. These
Fig. ”

“ I suppose 33 must mean something in Geometry; what does it mean?” “Nothing at all,” replied I, “not at least in Geometry; for Geometry has only Two Dimensions.” And then I began to shew the boy how a Point by moving through a length of three inches makes a Line of three inches, which may be represented by 3; and how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches every way, which may be represented by 32. ”

“ In all tests made with straight edges, the straight edge should be turned end for end and each measurement repeated, because, if the straight edge is true, turning it end for end will make no difference to the measurement, while if the straight edge is not true the measurement will vary when the straight edge is reversed. ”

“ To a Lineate belongeth an Angle and a Figure.
The common affections of a Magnitude were to be bounded, cutt, jointly measured, and adscribed: Then of a line to be right, crooked, touch'd, turn'd about, and [22] wreathed: All which are in a lineate by meanes of a line. Now the common affections of a Lineate are to bee Angled and Figured. And surely an Angle and a figure in all Geometricall businesses doe fill almost both sides of the leafe. And therefore both of them are diligently to be considered.
3. An Angle is a lineate in the common section of the bounds. ”

“ Neither may a surface, of 2. dimensions, be bounded with [23] one right line: Nor a body, of three dimensions, bee bounded with two, at lest beeing plaine surfaces.
4. The shankes of an angle are the bounds compreding the angle.
Scèle or Crura, the Shankes, Legges, H. are the bounds insisting or standing upon the base of the angle, which in the Isosceles only or Equicrurall triangle are so named of Euclide, otherwise he nameth them Latera, sides. So in the examples aforesaid, ea. and ei. are the shankes of the superficiary angle e ”