## The Mathematical Papers of Isaac Newton:, Volume 5; Volumes 1683-1684The fifth volume of this definitive edition centres around Newton's Lucasian lectures on algebra, purportedly delivered during 1673-83, and subsequently prepared for publication under the title Arithmetica Universalis many years later. Dr Whiteside first reproduces the text of the lectures deposited by Newton in the Cambridge University Library about 1684. In these much reworked, not quite finished, professional lectiones, Newton builds upon his earlier studies of the fundamentals of algebra and its application to the theory and construction of equations, developing new techniques for the factorizing of algebraic quantities and the delimitation of bounds to the number and location of roots, with a wealth of worked arithmetical, geometrical, mechanical and astronomical problems. An historical introduction traces what is known of the background to the parent manuscript and assesses the subsequent impact of the edition prepared by Whiston about 1705 and the revised version published by Newton himself in 1722. A number of minor worksheets, preliminary drafts and later augmentations buttress this primary text, throwing light upon its development and the essential untrustworthiness of its imposed marginal chronology. |

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### Contents

between given quadratic cubic and quartic equations two by two 126 Elimination | 4 |

his concept of a universal arithmetick mirroring | 15 |

Whiston V preface to the 1707 editio princeps Certain exaggerations stemming | 21 |

that of an Apollonius circle 230 Problems 278 | 27 |

resolving a quartic algebraic polynomial into surd quadratic factors 29 His concluding | 30 |

5r6v Delimitation of the roots of equations and their factoriza | 42 |

the analysis despite minor differences structurally the same as Descartes 270 | 46 |

23r Newtons derivation of theHeronian formula for a triangles | 50 |

construction of a parabola through four points by determining its Cartesian equation | 306 |

where two globes simultaneously released fall under simple | 332 |

possibility of multiple real roots 338 An equation can have as many roots as it | 350 |

diminishing them by a given quantity 352 Eliminating first or second terms from | 366 |

Determining rational roots of a numerical equation by finding its factors 368 Isolating | 372 |

Exemplified in equations of fourth fifth and seventh degree 376 The technique can | 384 |

computation of an equations root is easy geometrical constructions for effecting | 424 |

complicated algebraic curves 426 On this basis Newton urges the superiority of | 434 |

PART 2 | 131 |

two arith | 142 |

script | 156 |

metrical relationships into algebraic equivalents 158 The same derived algebraic equation | 162 |

Euclidean propositions useful to this end 164 Their application to Schootens problem | 168 |

Reducing the algebraic equivalent to a single equation 172 Different | 182 |

algebraic determi | 192 |

Schootens fishpond promenade 194 Problem 10 to cut off a given area | 204 |

the Newtonian geometrical model for anglesection and the algebraic | 212 |

the full cubic 438 Proof of the method 440 An equivalent circular neusis used to construct | 448 |

The circular neusis used to trisect an angle 45860 The equivalent rectilinear neusis | 462 |

braically higher curves 474 The parabola is geometrically less simple than the ellipse | 488 |

2V Two draft outlines of Newtons | 518 |

THE ARITHMETICS UNIVERSALIS LIBER PRIMUS | 533 |

which ensues in algebraic reduction from choosing as base variable one related in | 614 |

the reduced Apollonian circletangency problem 61618 Variant ways | 620 |

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### Common terms and phrases

1707 editio princeps Acta Eruditorum adeo adeoqj aequatio aequationis affirmativae algebraic angle anguli angulum Arithmetica Arithmetica Universalis Atqj aufer autem circle construction cujus datam datis Descartes differentia dimensionum divided dividendo divisor divisorem editio princeps ejus ellipse equal equation ergo erit etiam evadit extracted figure geometrical given in position globe globi habebitur hence hyperbola inter invenire itaqj latera latus rectum Lect Leibniz let fall linea linearum modo multiplied negativae nempe Newton nihil numeri omnes orietur parabola perpendicular positione potest Prob problem problematum propter punctis punctum quadratum quae quam quantitatis quantity Quare quartic equation quatuor quibus quod quotient quoties radicem radices radix ratio recta reduction reductione reductionem restabit root sides sive square straight line summa sunt termini terminorum triangle trianguli vero Whiston