Logic Is The Foundation Of The Certainty

Now I Will Have Less Distraction.

After Exponential Quantities The Circular Functions, Sine And Cosine, Should Be Considered Because They Arise When Imaginary Quantities Are Involved In The Exponential.

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.

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.

Although To Penetrate Into The Intimate Mysteries Of Nature And Thence To Learn The True Causes Of Phenomena Is Not Allowed To Us, Nevertheless It Can Happen That A Certain Fictive Hypothesis May Suffice For Explaining Many Phenomena.