Line (geometry)

“ 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? ”

“ 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 ”

“ 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. ”

“ There are three famous problems which are now admitted to be beyond the power of Geometry; namely, to find a straight line equal in length to the circumference of a given circle, to trisect any given angle, and to find two mean proportionals between two given straight lines. The grounds on which the geometrical solution of these problems is admitted to be impossible cannot be explained without a knowledge of the higher parts of mathematics ”

“ 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. ”

“ The principle of § 16, that the algebraical sum of the projections of the sides of any closed polygon on any line is zero, or that the two sets of sides of the polygon which connect a vertex A and a vertex B have the same sum of projections on the line, in sign and magnitude, as we pass from A to B, is applicable when the sides do not all lie in one plane. ”

“ It is perfectly possible to set, up a one-to-one correspondence between the points of a line and the points of a plane, or, indeed, between the points of a line and the points of a space of any finite number of dimensions, if the correspondence is not restricted to be continuous. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ This last equation, of which the left-hand side satisfies the condition for breaking up into two factors, represents the two imaginary circular points at infinity, through which all circles and their asymptotes pass.
There is strict duality in descriptive geometry between point-line-locus and line-point-envelope theorems. But in metrical geometry duality is encumbered by the fact that there is in a plane one special line only, associated with distance, while of special points, associated with direction, there are two: moreover the line is real, and the points both imaginary. ”

“ A plane perpendicular to a radius at its extremity is tangent to the sphere.
The student should always have his attention called to the analogue in plane geometry, where there is one. If here we pass a plane through the radius in question, the figure formed on the plane will be that of a line tangent to a circle. If we revolve this about the line of the radius in question, as an axis, the circle will generate the sphere again, and the tangent line will generate the tangent plane. ”

“ A few illustrations, as of an open book, the wall and floor of a room, and a swinging door, serve to make the concept clear, while a plane at right angles to the edge shows the measuring plane angle. So manifest is this [Pg 299] relationship between the dihedral angle and its measuring plane angle that some teachers omit the proposition that two dihedral angles have the same ratio as their plane angles.
Theorem. If two planes are perpendicular to each other, a straight line drawn in one of them perpendicular to their intersection is perpendicular to the other. ”

“ 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. ”

“ 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. ”

“ The student of modern geometry is already accustomed to think of parallel lines as lines which intersect at infinity, or as lines whose inclination is zero. In speaking of the generalized geometry in a plane, we may define that two straight lines which intersect upon the fundamental conic are parallel. It thus follows that through any point two distinct parallels can be drawn to a given straight line. The only exception will arise in the case where the given line touches the fundamental conic. ”

“ Geometrical lines and figures are properly nothing more than numbers, or the fundamental arithmetical notions fixed in space and invested with a corporeal shape, and thereby become visible. It is, however, not unusual to regard the first principles of geometry—such as the point, the right line, the square, and the triangle, out of which all else is compounded—as independent of experience, and existing absolutely in and by themselves. ”

“ 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. ”

“ 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. ”

“ Given the middle point of a line segment, to draw a line parallel to the segment and passing through a given point.
6. A line is drawn cutting the sides of a triangle ABC in the points A', B', C' the point A' lying on the side BC, etc. The harmonic conjugate of A' with respect to B and C is then constructed and called A". Similarly, B" and C" are constructed. Show that A"B"C" lie on a straight line. Find other sets of three points on a line in the figure. Find also sets of three lines through a point.
[pg 29] CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS
Fig. ”

“ In solid geometry the elementary figures are the point, the line and the plane; we have, moreover, first, that which under one aspect is the curve and under another aspect the developable (or torse) , and which may be regarded as a singly infinite system of points, of lines or of planes; and secondly, the surface, which may be regarded as a doubly infinite system of points or of planes, and also as a special triply infinite system of lines. (The tangent lines of a surface are a special complex.) ”

“ Circles and Spheares are equall, which have equall diameters. For the raies, which doe measure the space betweene the Center and Perimeter, are equall, of which, being doubled, the Diameter doth consist. Sn.
[51]
The fifth Booke, of Ramus his Geometry, which is of Lines and Angles in a plaine Surface.
1. A lineate is either a Surface or a Body.
Lineatum, (or Lineamentum) a magnitude made of lines, as was defined at 1. e. iij. is here divided into two kindes: which is easily conceived out of the said definition there, in which a line is excluded, and a Surface & a body are comprehended. ”

“ Because, by the grant, the right line ae, is cut proportionally, therefore the Oblong of the secant and outter segment, is equall to the quadrate of the greater segment, to which by the grant, the base ei, is equall. Here therefore the angle aie is the double of the angle at a: because it is equall to the angles aio, and oai, which are equall betweene themselves. For by the 27 e xvj it is equall to the angle oai in the alterne segment. And the remainder aio, is equall to it selfe. Therefore also the angle aei, is equall to the same two angles, because it is equall to the angle aie. ”

“ Yet if any man shall desire a demonstration of it, it is thus: Let therefore the diagramme or figure bee in the triangles aei, and aou: And the angles oau, and eai, let them be equall: And as ua is to ae, so let ia be unto ao: I say that the triangles aou, and eai, are equall. For eo being knit together with a right line, uao is unto oae, as ua is unto ae, by the 7 e: [101] And ia, unto ao, by the grant, is as eai is unto eao. Therefore uao, and eai, are unto eao proportionall: And therefore they are equall one to another. ”

“ 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. ”

“ Now a line lyeth equally betweene his owne bounds, when it is not here lower, nor there higher: But is equall to the space comprehended betweene the two bounds or ends: As here ae. is, so hee that maketh rectum iter, a journey in a straight line, commonly he is said to treade so much ground, as he needes must, and no more: He goeth obliquum iter, a crooked way, which goeth more then he needeth, as Proclus saith.
[15]
6. A right line is the shortest betweene the same bounds. ”

“ The possibility of a right line is a square H. A right line is said posse quadratum, to be in power a square; because being multiplied in it selfe, it doth make a square.
5 If two conterminall perpendicular equall right lines be closed with parallells, they shall make a quadrate. ”

“ Draw a periphery by three points, which doe not fall in a right line.
As here, by aei, (First from a, to e, let a right line be drawne; And likewise from e to i. Then, by the 12 e v, let both these lines be cut into equall parts, by two infinite right lines: These halfing lines also shall meete: And in their meeting shall be the Center, by the 8 e. And therefore from that meeting unto any of the sayd points given is the ray of the periphery desired.)
10. If a diameter doe halfe an inscript, that is, not a diameter, it doth cut it perpendicularly: And contrariwise: 3 p iij. ”

“ So long as we agree that beginners in Geometry shall be limited to the use of Lines and Circles, so long will it be unsafe to assume a point as found, or a Line as drawn, merely because we are sure it exists. For example, it is axiomatic, of course, that every angle has a bisector: but it is equally obvious that it has two trisectors: and if I may assume the one as drawn, why not the others also? However we have discussed this matter already (p. 20) .
Min. A second method is 'superposition,' adopted by Mr. Wilson and Mr. ”