Selected quad for the lemma: power_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
power_n	let_v	line_n	rational_a	4,990	5	12.3256	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A05115	Via regia ad geometriam. = The vvay to geometry Being necessary and usefull, for astronomers. Geographers. Land-meaters. Sea-men. Engineres. Architecks. Carpenters. Paynters. Carvers, &c. Written in Latine by Peter Ramus, and now translated and much enlarged by the learned Mr. William Bedvvell.; Via regia ad geometriam. English	Ramus, Petrus, 1515-1572.; Bedwell, William, ca. 1561-1632.; Clarke, John, d. 1658.	1636 (1636)	STC 15251; ESTC S108337	93,096	205

in a periphery and doe differ onely in base 14 The angles in opposite sections are equall to two right angles 22. p iij. The reason or rate of a section is thus The similitude doth follow 15 If sections doe receive or containe equall angles they are alike e 10. d iij. 16 If like sections be upon an equall base they are equall and contrariwise 23,24 p iij. In the first figure let the base be the same And if they shall be said to unequall sections and one of them greater than another the angle in that a o e shall be lesse than the angle a i e in the lesser section by the 16 e vj. which notwithstanding by the grant is equall In the second figure if one section be put upon another it will agree with it Otherwise against the first part like sections upon the same base should not be equall But congruency is here sufficient By the former two propositions and by the 9 e x v. one may finde a section like unto another assigned or else from a circle given to cut off one like unto it 17 An angle of a section is that which is comprehended of the bounds of a section 18 A section is either a semicircle or that which is unequall to a semicircle A section is two fold a semicircle to wit when it is cut by the diameter or unequall to a semicircle when it is cut by a line lesser than the diameter 19 A semicircle is the halfe section of a circle Or it is that which is made the diameter Therefore 20 A semicircle is comprehended of a periphery and the diameter 18 dj 21 The angle in a semicircle is a right angle The angle of a semicircle is lesser than a rectilineall right angle But greater than any acute angle The angle in a greater section is lesser than a right angle Of a greater it is a greater In a lesser it is greater Of a lesser it is lesser ê 31 and 16. p iij. Or thus The angle in a semicircle is a right angle the angle of a semicircle is lesse than a right rightlined angle but greater than any acute angle The angle in the greater section is lesse than a right angle the angle of the greater section is greater than a right angle the angle in the lesser section is greater than a right angle the angle of the lesser section is lesser than a right angle H. The second part That the angle of a semicircle is lesser than a right angle is manifest out of that because it is the part of a right angle For the angle of the semicircle a i e is a part of the rectilineall right angle a i u. The third part That it is greater than any acute angle is manifest out of the 23. e x v. For otherwise a tangent were not on the same part one onely and no more The fourth part is thus made manifest The angle at i in the greater section a e i is lesser than a right angle because it is in the same triangle a e i which at a is right angle And if neither of the shankes be by the center notwithstanding an angle may be made equall to the assigned in the same section The fifth is thus The angle of the greater section e a i is greater than a right angle because it containeth a right-angle The sixth is thus the angle a o e in a lesser section is greater than a right angle by the 14 e x v j. Because that which is in the opposite section is lesser than a right angle The seventh is thus The angle e a o is lesser than a right-angle Because it is part of a right angle to wit of the outter angle if i a be drawne out at length And thus much of the angles of a circle of all which the most effectuall and of greater power and use is the angle in a semicircle and therefore it is not without cause so often mentioned of Aristotle This Geometry therefore of Aristotle let us somewhat more fully open and declare For from hence doe arise many things Therefore 22 If two right lines jointly bounded with the diameter of a circle be jointly bounded in the periphery they doe make a right angle Or thus If two right lines having the same termes with the diameter be joyned together in one point of the circomference they make a right angle H. This corollary is drawne out of the first part of the former Element where it was said that an angle in a semicircle is a right angle And 23 If an infinite right line be cut of a periphery of an externall center in a point assigned and contingent and the diameter be drawne from the contingent point a right line from the point assigned knitting it with the diameter shall be perpendicular unto the infinite line given Let the infinite right line be a e from whose point a a perpendicular is to be raised And 24 If a right line from a point given making an acute angle with an infinite line be made the diameter of a periphery cutting the infinite a right line from the point assigned knitting the segment shall be perpendicular upon the infinite line As in the same example having an externall point given let a perpendicular unto the infinite right line a e be sought Let the right line i o e be made the diameter of the peripherie and withall let it make with the infinite right line giyen an acute angle in e from whose bisection for the center let a periphery cut the infinite c. And 25 If of two right lines the greater be made the diameter of a circle and the lesser jointly bounded with the greater and inscribed be knit together the power of the greater shall be more than the power of the lesser by the quadrate of that which knitteth them both together ad 13 p. x. 26 If a right line continued or continually made of two right lines given be made the diameter of a circle the perpendicular from the point of their continuation unto the periphery shall be the meane proportionall betweene the two lines given 13 p vj. So if the side of a quadrate of 10. foote content were sought let the sides 1 foote and 10 foote an oblong equall to that same quadrate be continued the meane proportionall shall be the side of the quadrate that is the power of it shall be 10. foote The reason of the angles in opposite sections doth follow 27 The angles in opposite sections are equall in the alterne angles made of the secant and touch line 32. p iij. As let the unequall sections be e i o and e a o the tangent let it be u e y And the angles in the opposite sections e a o and e i o. I say they are equall in the alterne angles of the secant and touch line o e y and o e u. First that which is at a is equall to the

quem Agricola alijex antiquis monumentis tradi derunt Now by any one of these knowne and compared with ours to all English men well knowne the rest may easily be proportioned out 2. The thing proposed to bee measured is a Magnitude Magnitudo a Magnitude or Bignesse is the subject about which Geometry is busied For every Art hath a proper subject about which it doth employ al his rules and precepts And by this especially they doe differ one from another So the subject of Grammar was speech of Logicke reason of Arithmeticke numbers and so now of Geometry it is a magnitude all whose kindes differences and affections are hereafter to be declared 3. A Magnitude is a continuall quantity A Magnitude is quantitas continua a continued or continuall quantity A number is quantitas discreta a disjoined quantity As one two three foure doe consist of one two three foure unities which are disjoyned and severed parts whereas the parts of a Line Surface and Body are contained and continued without any manner of disjunction separation or distinction at all as by and by shall better and more plainely appeare Therefore a Magnitude is here understood to be that whereby every thing to be measured is said to bee great As a Line from hence is said to be long a Surface broade a Body solid Wherefore Length Breadth and solidity are Magnitudes 4. That is continuum continuall whose parts are contained or held together by some common bound This definition of it selfe is somewhat obscure and to be understand onely in a geometricall sense And it dependeth especially of the common bounde For the parts which here are so called are nothing in the whole but in a potentia or powre Neither indeede may the whole magnitude bee conceived but as it is compact of his parts which notwithstanding wee may in all places assume or take as conteined and continued with a common bound which Aristotle nameth a Common limit but Euclide a Common section as in a line is a Point in a surface a Line in a body a Surface 5. A bound is the outmost of a Magnitude Terminus a Terme or Bound is here understood to bee that which doth either bound limite or end actu in deede as in the beginning and end of a magnitude Or potentia in powre or ability as when it is the common bound of the continuall magnitude Neither is the Bound a parte of the bounded magnitude For the thing bounding is one thing and the thing bounded is another For the Bound is one distance dimension or degree inferiour to the thing bounded A Point is the bound of a line and it is lesse then a line by one degree because it cannot bee divided which a line may A Line is the bound of a surface and it is also lesse then a surface by one distance or dimension because it is only length wheras a surface hath both length and breadth A Surface is the bound of a body and it is lesse likewise then it is by one dimension because it is onely length and breadth whereas as a body hath both length breadth and thickenesse Now every Magnitude actu in deede is terminate bounded and finite yet the geometer doth desire some time to have an infinite line granted him but no otherwise infinite or farther to bee drawane out then may serve his turne 6. A Magnitude is both infinitely made and continued and cut or divided by those things wherewith it is bounded A line a surface and a body are made gemetrically by the motion of a point line and surface Item they are conteined continued and cut or divided by a point line and surface But a Line is bounded by a point a surface by a line And a Body by a surface as afterward by their severall kindes shall be understood Now that all magnitudes are cut or divided by the same wherewith they are bounded is conceived out of the definition of Continuum e. 4. For if the common band to containe and couple together the parts of a Line surface Body be a Point Line and Surface it must needes bee that a section or division shall be made by those common bandes And that to bee dissolved which they did containe and knitt together 7. A point is an undivisible signe in a magnitude A Point as here it is defined is not naturall and to bee perceived by sense Because sense onely perceiveth that which is a body And if there be any thing lesse then other to be perceived by sense that is called a Point Wherefore a Point is no Magnitude But it is onely that which in a Magnitude is conceived and imagined to bee undivisible And although it be voide of all bignesse or Magnitude yet is it the beginning of all magnitudes the beginning I meane potentiâ in powre 8. Magnitudes commensurable are those which one and the same measure doth measure contrariwise Magnitudes incommensurable are those which the same measure cannot measure 1 2. d. X. Magnitudes compared betweene themselves in respect of numbers have Symmetry or commensurability and Reason or rationality Of themselves Congruity and Adscription But the measure of a magnitude is onely by supposition and at the discretion of the Geometer to take as pleaseth him whether an ynch an hand breadth foote or any other thing whatsoever for a measure Therefore two magnitudes the one a foote long the other two foote long are commensurable because the magnitude of one foote doth measure them both the first once the second twice But some magnitudes there are which have no common measure as the Diagony of a quadrate and his side 116. p. X. actu in deede are Asymmetra incommensurable And yet they are potentiâ by power symmetra commensurable to witt by their quadrates For the quadrate of the diagony is double to the quadrate of the side 9. Rationall Magnitudes are those whose reason may bee expressed by a number of the measure given Contrariwise they are irrationalls 5. d. X. Ratio Reason Rate or Rationality what it is our Authour and likewise Salignacus have taught us in the first Chapter of the second booke of their Arithmetickes Thither therefore I referre thee Data mensura a Measure given or assigned is of Euclide called Rhetè that is spoken or which may be uttered definite certaine to witt which may bee expressed by some number which is no other then that which as we said was called mensura famosa a knowne or famous measure Therefore Irrationall magnitudes on the contrary are understood to be such whose reason or rate may not bee expressed by a number or a measure assigned As the side of the side of a quadrate of 20. foote unto a magnitude of two foote of which kinde of magnitudes thirteene sorts are mentioned in the tenth booke of Euclides Elements such are the segments of a right line proportionally cutte unto the whole line The Diameter in a circle is rationall But it is irrationall unto the side of