## The Mathematician |

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Abſciſs Abſciſſa alſo Anſwered by John aſſigned Baſe becauſe biſe&t Caſe Center Circle circumſcribing conſequently conſidered Curve D E M o N S T R A T demonſtrated deſcribed Diameter Diſtance Dočtrine draw drawn Ellipſe equal Equation expreſſed Expreſſion finite firſt flowing Quantities Fluxions Geometry given greateſt Hyperbola increaſe infinite inſcribed Inſtant Interſe&tion John Turner laſt leaſt leſs leſſer likewiſe Magnitude Method of Fluxions moſt Motion muſt N s T R A T Number Ordinate P R o B L E M Parabola parallel Parallelogram Parameter paſſing perpendicular Point Pos I T Poſition PR o Pos Problem Progreſſion Prop Proportion propoſed Propoſition Radius Reaſoning Rećtangle repreſent reſpectively right Angle right Line ſaid ſame ſecond ſhall ſhew Sides ſimilar Triangles ſince Sine ſome Space Square ſubſtituted Subtangent ſuch ſuppoſed Tangent theſe thoſe tranſverſe Axe ultimate Ratio Uſe vaniſh Velocity Vertex whence whoſe

### Popular passages

Page 157 - C' (89) (90) (91) (92) (93) 112. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference.

Page 189 - Velocity with which they increase and are generated; I sought a Method of determining Quantities from the Velocities of the Motions or Increments, with which they are generated; and calling these Velocities of the Motions or Increments Fluxions, and the generated Quantities Fluents, I fell by degrees upon the Method of Fluxions, which I have made use of here in the Quadrature of Curves, in the Years 1665 and 1666.

Page 6 - They found, that similar triangles are to each other in the duplicate ratio of their homologous sides; and, by resolving similar polygons into similar triangles, the same proposition was extended to these polygons also.

Page 13 - ... all their theorems of this kind. It is often said, that curve lines have been considered by them as polygons of an infinite number of sides. But this principle no where appears in their writings. We never find them resolving any figure, or solid, into infinitely small elements.

Page 57 - Whatfoever politive ideas we have in our minds of any fpace, duration, or number, let them be ever fo great, they are ftill finite ; but when we fuppofe an inexhauftible remainder, from which we remove all bounds, and wherein we allow the mind an endlefs...

Page 201 - ... time approach to each other within less than any given difference, become ultimately equal. If you deny it, let them be ultimately unequal, and let their ultimate difference be D, then they cannot approach nearer to equality than quantities having a difference D: which is against the hypothesis.

Page 189 - I consider mathematical quantities in this place not as consisting of very small parts, but as described by a continued motion. Lines are described, and therefore generated not by the apposition of parts, but by the continued motion of points ; superficies by the motion of lines ; solids by the motion of superficies ; angles by the rotation of the sides ; portions of time by a continual flux ; and so in other quantities. These geneses really take place in the nature of things, and are daily seen...

Page 65 - ... of the simple addition of rising Moments, or of the continual flux of one Moment, and for that reason ascribe only length to it, and determine its quantity by the length of the line passed over : As a line, I say, is looked upon to be the trace of a point moving forward, being in some sort divisible by a point, and may be divided by Motion one way, viz. as to length ; so Time may be conceived as the trace of a Moment continually flowing, having some kind of divisibility from an Instant, and from...

Page 188 - ... flowing quantities." For example: I don't here consider Mathematical Quantities as composed of Parts extremely small, but as generated by a continual motion. Lines are described, and by describing are generated, not by any apposition of Parts, but by a continual motion of Points. Surfaces are generated by the motion of Lines, Solids by the motion of Surfaces, Angles by the Rotation of their Legs, Time by a continual flux...

Page 6 - ... objections that have been made to it. But, before we proceed, it may be of use to consider the...