An Expository Sketch of a New Theory of the Calculus |
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Page 2
William Batchelder Greene. first ; and when an increase in the value of the ... values , y takes also , and by necessity , various continuous values that are ... special point , forwards , backwards or to one side . ( An attempt to ...
William Batchelder Greene. first ; and when an increase in the value of the ... values , y takes also , and by necessity , various continuous values that are ... special point , forwards , backwards or to one side . ( An attempt to ...
Page 12
... special angle but the increment ( or , perhaps , decrement ) of the ordinate of the curve will vary regularly with ... value at the point whose abscissa is æ , which special value is identical with the constant value of the rate of ...
... special angle but the increment ( or , perhaps , decrement ) of the ordinate of the curve will vary regularly with ... value at the point whose abscissa is æ , which special value is identical with the constant value of the rate of ...
Page 14
... special ordinate , since its cor- responding abscissa has a special value , x , is at once the or- dinate of the line a'de , and the ordinate of the tangent that is drawn to a'de , at d . Moreover , Y , regarded as ordinate of the line ...
... special ordinate , since its cor- responding abscissa has a special value , x , is at once the or- dinate of the line a'de , and the ordinate of the tangent that is drawn to a'de , at d . Moreover , Y , regarded as ordinate of the line ...
Page 15
... special déter- dY minate value , a :) dx and the rate of increase of y is ( the abscissa receiving the same special determination ) dy We dx have , therefore , dY dx dy — n x ; dx which is equivalent to saying that , since Y is always n ...
... special déter- dY minate value , a :) dx and the rate of increase of y is ( the abscissa receiving the same special determination ) dy We dx have , therefore , dY dx dy — n x ; dx which is equivalent to saying that , since Y is always n ...
Page 17
... special ordinate , since its corresponding abscissa has a special value , x , is at once the ordinate of the line gh , and the ordinate of the tangent that is drawn to that line , at g . U has , therefore , a two- fold nature , one ...
... special ordinate , since its corresponding abscissa has a special value , x , is at once the ordinate of the line gh , and the ordinate of the tangent that is drawn to that line , at g . U has , therefore , a two- fold nature , one ...
Other editions - View all
An Expository Sketch of a New Theory of the Calculus William Batchelder Greene No preview available - 2009 |
An Expository Sketch of a New Theory of the Calculus (Classic Reprint) William Batchelder Greene No preview available - 2018 |
An Expository Sketch of a New Theory of the Calculus (Classic Reprint) William Batchelder Greene No preview available - 2017 |
Common terms and phrases
abscissa x analogy angle formed AVENUE DES CHAMPS-ÉLYSÉES axis of x becomes becomes x calculus centimetre common point concave conceive conception consequently contrary flexure contrary signs corresponding curvature curve whose equation curve y f(x cuts the axis decreasing derived line cuts differen differential triangle direction distance dy dx equa equal to unity equal to zero expression identical increasing function infinitesimal infinitesimal calculus Leibnitz line a'de line drawn parallel line gh linear logarithmic nate nature obtain ordi ordinate y particular point of contrary point of tangency point whose coordinates points of secancy positive primitive line rate of increase regarded right line drawn secant line second derived line space special determination special ordinate special rate special value straight line subtangent theorem tical tion TRANSCENDENTAL FUNCTIONS trigonometrical tangent values a little variable variation vary continuously
Popular passages
Page 75 - If, with a view to demonstrate any proposition, a certain point is supposed, by virtue of which certain other points are attained ; and such supposed point be itself afterwards destroyed or rejected by a contrary supposition ; in that case, all the other points attained thereby, and consequent thereupon, must also be destroyed and rejected, so as from thenceforward to be no more supposed or applied in the demonstration.
Page 34 - Hamlet with the part of the Prince of Denmark omitted, for so far I have said nothing whatever about technical education.
Page 10 - From (3) p b9 b tan0'= 2~b = 2Tb=2l but this is the trigonometrical tangent of the angle which the tangent to the curve at (a, b) makes with the axis of x. Hence the new axis of y ' is a tangent to the curve at the origm.
Page 85 - *=!* and by simply changing the axes, we have hence, the sub-tangent is equal to the modulus of the system of logarithms from which the curve is constructed. In the Naperian system M=l, and hence the sub-tangent will be equal to 1 = AE.
Page 29 - ... when referred to the same pole and initial line. Show that the radius of curvature at any point of this locus is J of the length of the radius vector to the corresponding point on the original curve. for a plane curve, the upper or lower sign being taken according as the curve is convex or concave to the axis of x, The normal at the point P of the conic Zabx = Ьхг + ay1 meets the axis of x at G.
Page 5 - ... Measures to the farm, the household, the mechanic arts, etc., are so extensive that we now present a distinct treatment of the subject. 333. These Practical Measurements include Measures of Surface, Measures of Volume, Measures of Capacity, and Comparison of Weights and of Money. MEASURES OF SURFACE. 334. A Surface is that which has length and breadth without thickness. THE EECTANGLE.
Page 56 - Further, it is to be known that it is man by means of whom the natural world is conjoined with the spiritual...
Page 48 - ... the three components (two horizontal and one vertical) of the movement under investigation. In what follows we shall consider only the horizontal movements; the same arguments, however, may be applied to the case of vertical movements. Let us suppose a small horizontal platform to be moving parallel to the axis of X, and let it be required to determine the acceleration of this motion. On this platform an apparatus is mounted which can oscillate parallel to the same axis with a natural period,...
Page 1 - A more general definition of function is : // two or more variables are so related that to any given set of values comprising one and but one value for each of the variables except one of them there corresponds at least one value of the latter variable, each of the variables is said to be a function of all the others. Thus, if v be the volume of a parallelepiped, say a slab, and if /, b and / be respectively its length, breadth, and thickness, each of the four variables is a function of the remaining...
Page 84 - The value m, which depends upon the value of the base a, of the system of logarithms employed, is called the modulus of the system.