Now I Will Have Less

If A Nonnegative Quantity Was So Small That It Is Smaller Than Any Given One, Then It Certainly Could Not Be Anything But Zero. To Those Who Ask What The Infinitely Small Quantity In Mathematics Is, We Answer That It Is Actually Zero. Hence There Are Not So Many Mysteries Hidden In This Concept As They Are Usually Believed To Be. These Supposed Mysteries Have Rendered The Calculus Of The Infinitely Small Quite Suspect To Many People. Those Doubts That Remain We Shall Thoroughly Remove In The Following Pages, Where We Shall Explain This Calculus.

Notable Enough, However, Are The Controversies Over The Series 1 - 1 + 1 - 1 + 1 - ... Whose Sum Was Given By Leibniz As 1/2, Although Others Disagree. ... Understanding Of This Question Is To Be Sought In The Word "sum"; This Idea, If Thus Conceived - Namely, The Sum Of A Series Is Said To Be That Quantity To Which It Is Brought Closer As More Terms Of The Series Are Taken - Has Relevance Only For Convergent Series, And We Should In General Give Up The Idea Of Sum For Divergent Series.

To Those Who Ask What The Infinitely Small Quantity In Mathematics Is, We Answer That It Is Actually Zero. Hence There Are Not So Many Mysteries Hidden In This Concept As They Are Usually Believed To Be.

The Kind Of Knowledge Which Is Supported Only By Observations And Is Not Yet Proved Must Be Carefully Distinguished From The Truth; It Is Gained By Induction, As We Usually Say. Yet We Have Seen Cases In Which Mere Induction Led To Error. Therefore, We Should Take Great Care Not To Accept As True Such Properties Of The Numbers Which We Have Discovered By Observation And Which Are Supported By Induction Alone. Indeed, We Should Use Such A Discovery As An Opportunity To Investigate More Exactly The Properties Discovered And To Prove Or Disprove Them; In Both Cases We May Learn Something Useful.

A Function Of A Variable Quantity Is An Analytic Expression Composed In Any Way Whatsoever Of The Variable Quantity And Numbers Or Constant Quantities.