Elements of analytical geometry and the differential and integral calculus . called the axisof X, and the vcBtical line the axis of y, and they are markedas in the figure. The point A in the adjoining fig-ure is the zero point. Draw any lineas LL through this point, and de-signate the natural tangent of theangle LAX by a, (the radius beingunity.) Then take any distance on AX asAF, and represent it by x, and theperpendicular distance PJSf put equalto by trigonometry we have Rad. : tan. MAP i :; AF : PM\\.a.\.x\y ox (1) Now this-equation is general; that is, it applies to any poi
Elements of analytical geometry and the differential and integral calculus . called the axisof X, and the vcBtical line the axis of y, and they are markedas in the figure. The point A in the adjoining fig-ure is the zero point. Draw any lineas LL through this point, and de-signate the natural tangent of theangle LAX by a, (the radius beingunity.) Then take any distance on AX asAF, and represent it by x, and theperpendicular distance PJSf put equalto by trigonometry we have Rad. : tan. MAP i :; AF : PM\\.a.\.x\y ox (1) Now this-equation is general; that is, it applies to any pointM on the line AL, because we can make aj greater or less, andPJf will be greater or less in like proportion, and J!f will movealong on the line AL as we move P on the line AX. Becausethe point JSfwill continue on the line -4Athrough all changes of a?and y, we say that y=ax, is the equation of the line AL. Now let us diminish x to 0, and the equation reduces to y=0in the same time, which brings MonAo the point A. Let x pass the line YY\ it then, becomes —x. AP and the. STRAIGHT LINES. U con esponding value of y will be PM\ and being below the lineXX will therefore be minus. Therefore ±y=±aiP is the general equation of the line LL, extending indefinitely ineither direction. If the tangent a becomes less, the line will incline more to-wards the line XX. When a=0 the line will coincide withXXf when infinite, it will coincide with YV Now let AF be +x, and a become —a, then PM willcorrespond to y, and becomes mimis y, because it is below theaxis XX. Or, algebraically y=—ax, indicating some point Mbelow the horizontal axis. Now we think it has been shown that y=ax may represent anyline asZZ passing through A from the \st into the Zd quadrant,and y=—ax may be made to represent any line as UL passingthrough A from the 2d into the 4th quadrant. Therefore y= may he vnade to represent any straight line passing through the zeropoint. In case we have —a and —x, that is,
Size: 1609px × 1554px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookauthorrobinson, bookcentury1800, bookdecade1850, bookyear1856
