Transcendental [numbers], They Transcend The Power Of

Transcendental [numbers], They Transcend The Power Of Algebraic Methods.

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Since The Fabric Of The Universe Is Most Perfect And The Work Of A Most Wise Creator, Nothing At All Takes Place In The Universe In Which Some Rule Of Maximum Or Minimum Does Not Appear ... There Is Absolutely No Doubt That Every Affect In The Universe Can Be Explained Satisfactorily From Final Causes, By The Aid Of The Method Of Maxima And Minima, As It Can Be From The Effective Causes Themselves ... Of Course, When The Effective Causes Are Too Obscure, But The Final Causes Are Readily Ascertained, The Problem Is Commonly Solved By The Indirect Method.

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.

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.