OF SYMBOLS OF QUANTITY

Excerpts

• OF SYMBOLS OF QUANTITY

  In Euclid an angle is not defined as a magnitude but as the inclination of two lines, which never exceeds two right angles: and in most of the propositions in Euclid it is not necessary to treat an angle otherwise than as a change of direction of a line.

• OF SYMBOLS OF QUANTITY

  So also in Trigonometry, where angular magnitude in general is treated numerically, it is desirable to use the term angle for the sum of a number of angles, which may be greater than two or than any number of right angles.

• OF SYMBOLS OF QUANTITY

  For instance, if a distance $a$ miles be measured from a fixed point $O$ along (or parallel to) a given line in a standard direction, say east, to $A$; and a line $AB$ be cut off from $OA$ by measuring from $A$ in the opposite direction, or westward, a distance $b$ miles, the distance $OB$ may be said to be $= (a - b)$ miles east of $O$ not only in the case where $a > b$, but also when $a < b$, if it be agreed to interpret the result as meaning $a - b$ east of $O$ (or in the standard direction) if $a - b$ is a $+$ number, and $b - a$ miles west of $O$ (or in the contrary direction), when $a - b$ is a $-$ number.

• OF SYMBOLS OF QUANTITY

  Then the standard direction may be called the $+$ (or positive) direction, and the contrary direction the $-$ (or negative) direction.

• OF SYMBOLS OF QUANTITY

  Again, with reference to angles, if the hand of a going clock be put back through an angle $\theta$, then, after the time during which the hand moves through an angle $\phi$, the hand will make an angle $\theta - \phi$ with its present position, the angle being $+$ and measured in the opposite way to that in which the hand of the clock moves, if $\theta > \phi$; or $-$ and measured in the contrary direction, if $\theta < \phi$.

• OF SYMBOLS OF QUANTITY

  In what follows, an angle will be considered as if produced by the revolution of a radius of a circle, the direction of revolution from an initiatory position being considered as $-$ or $+$ according as it takes place in the direction of the motion of the hand of a clock or the reverse.

Equations

• OF SYMBOLS OF QUANTITY

  = (a - b)

  The signed position of B relative to O equals a - b miles, read as eastward when a - b is positive and westward when it is negative, so one formula covers both cases of a > b and a < b.

Problems

No exercises in this chapter.