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By Steven B. Engelsman

This e-book presents an in depth description of the most episodes within the emergence of partial differentiation throughout the interval 1690-1740. It argues that the advance of this idea - to a substantial measure of perfection - came about virtually completely in difficulties touching on households of curves. hence, the booklet exhibits the origins of the information and methods which lead the way for the surprising advent of partial differential equations in 1750. the most methodological attribute of the booklet is its emphasis on an entire realizing of the explanations, difficulties and objectives of the mathematicians of that point.

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Hence t h e d i f f i c u l t i e s which Johann B e r n o u l l i met i n c o n s t r u c t i o n s by “ q u a d r a t u r e s of c u r v e s ” r e l a t e t o t h e a n a l y t i c a l problem o f d i f f e r e n t i a t i n g an i n t e g r a l with r e s p e c t t o a parameter o c c u r r i n g under t h e i n t e g r a l s i g n . Johann B e r n o u l l i i l l u s t r a t e d h i s c r i t i c i s m of L e i b n i z ’ s remarks by s e t t i n g t h e t a n g e n t problem f o r e q u a l a r c s t r a j e c t o r i e s i n a f a m i l y of e l l i p s e s having, a common h o r i z o n t a l a x i s and v a r i a b l e v e r t i c a l a x e s .

I n f a c t , t h e r e i s q u i t e a paradox between L e i b n i z ' s c a r e - 25 Envelopes f u l geometric i n t r o d u c t i o n of f a m i l i e s of curves by means of t h e o r d i n a t r i x and t h e almost complete absence of t h e o r d i n a t r i x a s soon a s t h e a p p l i c a t i o n of t h e d i f f e r e n t i a l c a l c u l u s t o e q u a t i o n s of f a m i l i e s of c u r v e s i s e x p l a i n e d . A seemingly u n r e l a t e d d i s c u s s i o n then follows about e q u a t i o n s which r e p r e s e n t f a m i l i e s of c u r v e s .

2). In each point of the mirror one can by means of the law of reflection construct the reflected light ray. Now the family of these reflected light rays is an ordered family of straight lines, and the order in this family is derived from the order of the points on the mirror. Families of Curves in the 1690s 24 fig. 2 Thus the m i r r o r a c t s a s the o r d i n a t r i x of t h e f a m i l y of r e f l e c t e d l i g h t r a y s . "' G e n e r a l i s i n g from s t r a i g h t l i n e s t o c u r v e s , L e i b n i z a r r i v e d a t h i s concept of " i n f i n i t e l y many curves g i v e n by p o s i t i o n i n o r d e r e d sequence": "But I accept a s "drawn i n o r d e r e d sequence" n o t only s t r a i g h t l i n e s b u t a l s o any c u r v e s , provided t h e law i s known a c c o r d i n g t o which i n a g i v e n point of some given curve ( t h e o r d i n a t r i x ) t h e [curved] l i n e corresponding t o t h a t p o i n t can be drawn, which [ l i n e ] w i l l be one of t h o s e t h a t a r e t o be drawn i n ordered sequence, o r t h a t are given by p o s i t i o n .

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