## The Early Mathematical Manuscripts of Leibniz: Translated from the Latin Texts Published by Carl Immanuel Gerhardt with Critical and Historical Notes |

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already analysis angle appear applied arithmetical assumed axis Barrow base calculus called Cavalieri center of gravity characteristic circle connection considered constant contained corresponding curve Descartes diagram differences differential drawn equal equation essay evident example expressed fact figure follows further geometry Gerhardt give given Gregory Hence Huygens idea infinitely integration kind known later Leibniz less letter manner manuscript matter means mentioned method moments namely nature Newton notation obtained ordinates original parallel Pascal perpendicular plane possible powers probably problem produced progression proof proved published quadrature quantities ratio reason rectangle reduced referred regard remark result roots rule seems seen side similar solid solution square straight line suggestion suppose surface taken tangents theorem things tion triangle Wallis whole writing written

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Page 22 - It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it.

Page 150 - ... ay, even though he think that such things are utterly impossible; it will be sufficient simply to make use of them as a tool that has advantages for the purpose of the calculation, just as the algebraists retain imaginary roots with great profit. For they contain a handy means of reckoning, as can manifestly be verified in every case in a rigorous manner by the method already stated. Thus Leibniz presents his calculus of infinitesimals as an abbreviated form of the rigorous Greek method of exhaustion,...

Page 136 - Elementa calculi novi pro differentiis et summis, tangentibus et quadraturis, maximis et minimis, dimensionibus linearum, superficierum, solidorum, aliisque communem calculum transcendentibus.

Page 212 - All these theorems are true for series in which the differences of the terms bear to the terms themselves a ratio which is less than any assignable quantity.

Page 215 - Cartesian algebra and also of the method of indivisibles,35 indeed I did not know the correct definition of the center of gravity. For, when by chance I spoke of it to Huygens, I let him know that I thought that a straight line drawn through the center of gravity always cut a figure into two equal parts ; since that clearly happened in the case of a square, or a circle, an ellipse, and other figures that have a center of magnitude, I imagined that it was the same for all other figures. Huygens laughed...

Page 180 - DE, continually 117 increase from the initial ordinate AZ; also let AIF be a line such that, if any straight line EDF is drawn perpendicular to AD, cutting the curves in the points E, F, and AD in D, the rectangle contained by DF and a given length R is equal to the intercepted space ADEZ; also let DE : DF = R : DT, and join DT. Then TF will touch the curve AIF.

Page 152 - ... quantities (ie, the very least of those within our knowledge), it is understood that we mean quantities that are indefinitely great or indefinitely small, ie, as great as you please, or as small as you please, so that the error that any one may assign may be less than a certain assigned quantity, On these suppositions, all the rules of our algorithm, as set out in the Acta Eruditorum for October 1684, can be proved without much trouble. Leibniz then goes over these rules.

Page 82 - But f means a sum, and da difference. From the given y, we can always find y/d, or ?, that is the difference of the y's.

Page 150 - Thus, by infinitely great and infinitely small, we understand something indefinitely great, or something indefinitely small, so that each conducts itself as a sort of class, and not merely as the last thing of a class.

Page 149 - ... a state that the difference is less than any assignable quantity; also that in this state there will still remain some difference, some velocity, some angle, but in each case one that is infinitely small . . . For the present, whether such a state of instantaneous transition from inequality to equality...