## The MathematicianJ. Wilcox, 1751 - 399 pages |

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**Demonstrations**, and fome of his Expreffions , ( as when he speaks of Quantities more than infinite ) have been ...**Demonstration**, which requires the Suppofition of no Quantities , but fuch as are finite , and easily conceived ; by ... Page 23

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**DEMONSTRATION**. Let HO - Y , GC = y , VH - X , and VG = x ; then ( by Prop . 1. ) pX = Y ' , and px - y ' ; there- fore , px - px - Y - y ' ; confequently , p : Y + y :: Y — y : X— * ; that is , p : IN : NO : NC . Q. E. D. PRO ... Page 24

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**DEMONSTRATION**. Let HSb , VG = x , VH = X , HO = Y , and 2 GM = y ; then ( by Prop . 1. ) y2 : Y1 :: x : X :: by fimilar Triangles , y : b ; therefore , b = Y1 = pX ; - y ་ or by = px ; that is , p : GM :: HS : HV . Q.E.D. PROPOSITION ... Page 25

**DEMONSTRATION**. 2 2 2 Let VDb , VP = x , VH = X ; then PD = b + x , and HD = b + X ; but from fimilar Trian- gles PD : HD : PCH ( by Prop . 1. ) PV HV ; that is , b + 2bx + x 2 : b 2 + 2 b X + XX ; therefore Xxxb2 - xxx xX ; or bxX ... Page 27

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**DEMONSTRATION**. Let QGb ; then ( by Eu . 8. 6. ) GT : GC :: GC : GQ ; that is , 2x : y :: y : b ; therefore , 2 bx = y = px ; whence bp , or QG = ÷ p . Q.E.D. PROPOSITION VIII . The Distance from the Focus to the Point of Contact ; from ...### Common terms and phrases

Abfciffa Abfcifs affigned Affymptotes alfo alſo Anfwered by John Arch Bafe Baſe becauſe bifecting Cafe Center Circle circumfcribing Cofine confequently Curve defcribed DEMONSTRATION determine Diameter Difference Diſtance draw Ellipfe equal Equation expreffed Expreffion faid fame fecond fhall fhew fimilar Triangles fince finite firft firſt flowing Quantities Fluent Fluxions fome fuch fuppofed Geometry given greateſt Hyperbola increaſe infcribed infinite infinite Series Interfection John Turner laft leffer lefs Magnitude Meaſure Method Method of Fluxions Motion muſt Number Ordinate paffing Parabola parallel Parallelogram Parameter perpendicular Pofition Point of Contact PROBLEM Progreffion Prop propofed Proportion Q. E. D. PROPOSITION Radius Reaſoning Rectangle refpectively reprefent right Angle right Line Semi-diameter Sides Sine Sir Ifaac Space Square Subtangent Tangent thefe theſe thofe thoſe thro tion tranfverfe Axe Trapezium ultimate Ratio Velocity Vertex whence whofe

### 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 193 - 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 205 - ... 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 193 - 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 192 - ... 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...