Selected quad for the lemma: power_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
power_n	line_n	number_n	square_a	3,745	5	14.0094	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A95751	The trissotetras: or, a most exquisite table for resolving all manner of triangles, whether plaine or sphericall, rectangular or obliquangular, with greater facility, then ever hitherto hath been practised: most necessary for all such as would attaine to the exact knowledge of fortification, dyaling, navigation, surveying, architecture, the art of shadowing, taking of heights, and distances, the use of both the globes, perspective, the skill of making the maps, the theory of the planets, the calculating of their motions, and of all other astronomicall computations whatsoever. Now lately invented, and perfected, explained, commented on, and with all possible brevity, and perspicuity, in the hiddest, and most re-searched mysteries, from the very first grounds of the science it selfe, proved, and convincingly demonstrated. / By Sir Thomas Urquhart of Cromartie Knight. Published for the benefit of those that are mathematically affected.	Urquhart, Thomas, Sir, 1611-1660.	1645 (1645)	Wing U140; Thomason E273_9; ESTC R212170	85,776	129

flat and blunt Angles Occurse is a meeting together from occurro occursum Oppobasall is said of those Moods which have a Catheteuretick Concordance in their Datas of the same Cathetopposite Angles and the same Bases Oppocathetall is said of those Loxogonosphericals which have a Datisterurgetick Concordance in their Datas of the same Angles at the Base and the Perpendicular Oppoverticall is said of those Moods which have a Catheteuretick Concordance in their Datas of the same Cathetopposites and verticall Angles Orthogonosphericall is said of right angled Sphericals of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 rectus 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 angulus and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 gobus Oxygonosphericall is said of acute-angled sphericals of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 P. PArallelisme is a Parallel equality of right lines cut with a right line or of Sphericals with a Sphericall from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 equidistans of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Parallelogram is an oblong long square rectangle or figure made of parallel lines of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 linea Partiall is said of enodandas depending on severall Axioms Particularise specialise by some especiall difference to contract the generality of a thing Partition is said of the severall operations of every Loxogonosphericall Mood and is divided in praenoscendall catheteuretick and hysterurgetick Permutat proportion or proportion by permutation or alternat proportion is when the Antecedent is compared to the Antecedent and the Consequent to the Consequent vide Perturbat Perpendicularity is the affection of the Perpendicular or plumb-line which comes from perpendendo id est explorando altitudinem Perturbat is the same as permutat and called so because the order of the Analogie is perturbed Planobliquangular is said of plaine Triangles wherein there is no right Angle at all Planorectangular is said of plaine right-angled Triangles Planotriangular is said of plaine Triangles that is such as are not Sphericall Pleuseotechnie the Art of Navigation of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 navigatio and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ars Plusminused is said of Moods which admit of Mensurators or whose illatitious termes are never the same but either more or lesse then the maine quaehtas Poliechyrologie the Art of fortifying Townes and Cities of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 urbs civit as 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 munio firmo and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ratio Possubservient is that which after another serveth for the resolving of a question of post and subserviens of sub and servio Potentia is that wherein the force and whole result of another thing lies Power is the square quadrat or product of a line extended upon it selfe or of a number in it selfe multiplied Powered squared quadrified Precept document from praecipio praeceptum Praeroscenda are the termes which must be knowne before we can attaine to the knowledge of the maine quaesitas of prae and nosco Praenoscendall is said of the Concordances of those Moods which agree in the same praenoscendas Praesection praesectionall is concerning the digit towards the left whose cutting off saveth the labour of subtracting the double or single Radius Praescinded problems are those speculative Datoquaeres which are not applied to any matter by way of practice Praesubservient is said of those Moods which in the first place we must make use of for the explanation of others of prae and ●ub●ervio Prime is said of the furthest Cathetopposite or Angle at the Base contained within the Triangle to be resolved Primifie the Radius is to put the Radius in the first place primumque inter terminos collocare proportionales Problem problemet a question or datoquaere from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 unde 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 propositum objectaculum Product is the result factus or operatum of a multiplication from produco productum Proportion proportionality are the same as Analogy and Analogisme the first being a likenesse of termes the other of proportions Proposition a proposed sentence whether theorem or problem Prosiliencie is a demission or falling of the Perpendicular from prosilio ex pro salio Proturgetick is said of the first operation of every Disergetick Mood of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 being Attically contracted into 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Q. QUadrant the fourth part of a Circle Quadranting the protracting of a Sphericall side unto a Quadrant Quadrat a Square a forma quadrae the power or possibility of a line vide Power Quadrobiquadraequation concerneth the Square of the subtendent side which is equall to the Biquadrat or two Squares of the Ambients Quadrosubduction is concerning the subtracting of the Square of one of the Ambients from the Square of the Subtendent Quaesitas the things demanded from quaero quaesitum Quotient is the result of a division from quoties how many times R. RAdically meeting is said of those Oblongs or Squares whose sides doe meet together Radius ray or beame is the Semidiameter called so metaphorically from the spoake of a wheele which is to the limb thereof as the Semidiameter to the circumference of a circle Reciprocall is said of proportionalities or two rowes of proportionals wherein the first of the first is to the first of the second as the last of the second is to the last of the first and contrarily Rectangular is said of those figures which have right Angles Refinedly is said when we go the shortest way to work by primifying the Radius Renvoy a remitting from one place to another it comes from the French word Renvoyer Representative is said of the letters which stand for whole words as E. for side L. for secant U. for subtendent Residuat is to leave a remainder nempe id quod residet superest Resolver is that which looseth and untieth the knot of a difficulty of re and solvo Resolutory is said of the last partition of the Loxogonosphericall operations Result is the last effect of a work Root is the side of a Square Cube or any cossick figure S. SCheme signifieth here the delineation of a Geometricall figure with all parts necessary for the illustrating of a demonstration from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 habeo Sciography the Art of shadowing of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 umbra and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 scribo Segment the portion of a thing cut off quasi secamentum quod a re aliqua secatur Sexagesimat subsexagesimat resubsexagesimat and biresubsexagesimat are said of the division subdivision resubdivision and reer-resubdivision of degrees into minuts seconds thirds and fourths in 60. of each other the devisor of the fore goer being successively the following dividend and the quotient alwayes sixty Sharp is said of acute Angles Sindiforall is said of those Moods the fourth terme of whose Analogie is onely illatitious to the maine quaesitum Sindiforation is the affection of those foresaid Moods whereby the value of

certain all Magnitudes being Figures at least in power and all Figures either Triangles or Triangled that the Arithmeticall Solution of any Geometricall question dependeth on the Doctrine of Triangles 5. And though the proportion betwixt the parts of a Triangle cannot be without some errour because of crooked lines to right lines and of crooked lines amongst themselves the reason is inscrutable no man being able to finde out the exact proportion of the Diameter to the Circumference yet both in plain Triangles where the measure of the Angles is of a different species from the sides and in Sphericalls wherein both the Angles and sides are of a circular nature crooked lines are in some measure reduced to right lines by the definition of quantity which right lines viz. Sines Tangents and Secants applyed to a Circle have in respect of the Radius o● half-Diameter 6. And therefore though the Circles Quadrature be not found out it being in our power to make the Diameter or the semi-Diameter which is the Radius of as many parts as we please and being sure so much the more that the Radius be taken the error will be the lesser for albeit the Sines Tangents and Secants be irrationall thereto for the most part and their proportion inexplicable by any number whatsoever whither whole or broken yet if they be rightly made they will be such as that in them all no number will be different from the truth by an integer or unity of those parts whereof the Radius is taken which is so exactly done by some especially by Petiscus who assumed a Radius of twenty six places that according to his supputation the Diameter of the Earth being known and the Globe thereof supposed to be perfectly round one should not fail in the dimension of its whole Circuit the nine hundreth thousand scantling of the Million part of an Inch and yet not be able for all that to measure it without amisse for so indivisible the truth of a thing is that come you never so neer it unlesse you hit upon it just to a point there is an errour still DEFINITIONS A Cord or Subtense is a right line drawn from the one extremity to the other of an Arch. 2. A right Sine is the half Cord of the double Arch proposed and from one extremity of the Arch falleth perpendicularly on the Radius passing by the other end thereof 3. A Tangent is a right line drawn from the Secant by one end of the Arch perpendicularly on the extremity of the Diameter passing by the other end of the said Arch. 4. A Secant is the prolonged Radius which passeth by the upper extremity of the Arch till it meet with the sine Tangent of the said Arch. 5. Complement is the difference betwixt the lesser Arch and a Quadrant or betwixt a right Angle and an Acute 6. The complement to a semi-Circle is the difference betwixt the half-Circumference and any Arch lesser or betwixt two right Angles and an Oblique Angle whither blunt or sharp 7. The versed sine is the remainder of the Radius the sine Complement being subtracted from it and though great use may be made of the versed sines for finding out of the Angles by the sides and sides by the Angles yet in Logarithmicall calculations they are altogether uselesse and therefore in my Trissotetras there is no mention made of them 8. In Amblygonosphericall● which admit both of an Extrinsecall and Intrinsecall demission of the perpendicular nineteen severall parts are to be considered viz. The Perpendicular the Subtendentall the Subtendentine two Cosubtendents the Basall the Basidion the chief Segment of the Base two Cobases the double Verticall the Verticall the Verticaline two Coverticalls the next Cathetopposite the prime Cathetopposite and the two Cocathetopposites fourteen whereof to wit the Subtendentall the Subtendentine the Cosubtendents the Basall the Basidion the Cobases the Verticall the Verticaline the Coverticalls and Cocathetopposites are called the first either Subtendent Base Topangle or Cocathetopposite whither in the great Triangle or the little or in the Correctangle if they be ingredients of that Rectangular whereof most parts are known which parts are alwayes a Subtendent and a Cathetopposite but if they be in the other Triangle they are called the second Subtendents Bases and so forth 9. The externall double Verticall is included by the Perpendicular and Subtendentall and divided by the Subtendentine the internall is included by cosubtendents and divided by the Perpendicular APODICTICKS THe Angles made by a right Line falling on another right Line are equall to two right Angles because every Angle being measured by an Arch or part of a Circumference and a right Angle by ninety Degrees if upon the middle of the ground line as Center be described a semi-Circle it will be the measure of the Angles comprehended betwixt the falling and sustaining lines 2. Hence it is that the four opposite Angles made by one line crossing another are always each to its own opposite equall for if upon the point of Intersection as Center be described a Circle every two of those Angles will fill up the semi-Circle therefore the first and second will be equall to the second and third and consequently the second which is the common Angle to both these couples being removed the first will remain equall to the third and by the same reason the second to the fourth which was to be demonstrated 3. If a right line falling upon two other right lines make the alternat Angles equall these lines must needs be Paralell for if they did meet the alternat Angles would not be equall because in all plain Triangles the outward Angle is greater then any of the remote inward Angles which is proved by the first 4. If one of the sides of a Triangle be produced the outward Angle is equall to both the inner and opposite Angles together because according to the acclining or declining of the conterminall side is left an Angulary space for the receiving of a paralell to the opposite side in the point of whose occourse at the base the Exterior Angle is divided into two which for their like and alternat situation with the two Interior Angles are equall each to its own conform to the nature of Angles made by a right line crossing divers paralells 5. From hence we gather that the three Angles of a plain Triangle are equall to two rights for the two inward being equall to the Externall one and there remaining of the three but one which was proved in the first Apodictick to be the Externall Angles complement to two rights it must needs fall forth what are equall to a third being equall amongst themselves that the three Angles of a plain Triangle are equall to two right Angles the which we undertook to prove 6. By the same reason the two acute of a Rectangled plain Triangle are equall to one right Angle and any one of them the others complement thereto 7.

In every Circle an Angle from the Center is two in the Limb both of them having one part of the Circumference for base for being an Externall Angle and consequently equall to both the Intrinsecall Angles and therefore equall to one another because of their being subtended by equall bases viz. the semi-Diameters it must needs be the double of the foresaid Angle in the limb 8. Triangles standing between two paralells upon one and the fame base are equall for the Identity of the base whereon they are seated together with the Equidistance of the Lines within the which they are confined maketh them of such a nature that how long so ever the line paralell to the base be protracted the Diagonall cutting of in one off the Triangles as much of bredth as it gains of length the ones losse accruing to the profit of the other Quantifies them both to an equality the thing we did intend to prove 9. Hence do we inferre that Triangles betwixt two paralells are in the same proportion with their bases 10. Therefore if in a Triangle be drawn a paralell to any of the sides it divideth the other sides through which it passeth proportionally for besides that it maketh the four segments to be four bases it becomes if two Diagonall lines be extended from the ends thereof to the ends of its paralell a common base to two equall Triangles to which two the Triangle of the first two segments having reference according to the difference of their bases and these two being equall as it is to the one so must it be to the other and therefore the first base must be to the second which are the Segments of one side of the Triangle as the third to the fourth which are the Segments of the second all which was to be demonstrated 11. From hence do we collect that Equiangled Triangles have their sides about the equall Angles proportionall to one another This sayes Petiscus is the golden Foundation and chief ground of Trigonometry 12. An Angle in a semi-Circle is right because it is equall to both the Angles at the base which by cutting the Diameter in two is perceivable to any 13. Of four proportionall lines the Rectangled figure made of the two extreames is equall to the Rectangular composed of the means for as four and one are equall to two and three by an Arithmeticall proportion and the fourth term Geometrically exceeding or being lesse then the third as the second is more or lesse then the first what the fourth hath or wanteth from and above the third is supplyed or impaired by the Surplusage or deficiency of the first from and above the second These Analogies being still taken in a Geometricall way make the oblong of the two middle equall to that of the extreams which was to be proved 14. In all plain Rectangled Triangles the Ambients are equall in power to the Subtendent for by demitting from the right Angle a Perpendicular there will arise two Correctangles from whose Equiangularity with the great Rectangle will proceed such a proportion amongst the Homologall sides of all the three that if you set them right in the rule beginning your Analogy at the main Subtendent seeing the including sides of the totall Rectangle prove Subtendents in the partiall Correctangles and the bases of those Rectanglets the Segments of the great Subtendent it will fall out that as the main Subtendent is to his base on either side for either of the legs of a Rectangled Triangle in reference to one another is both base and Perpendicular so the same bases which are Subtendents in the lesser Rectangles are to their bases the Segment of the prime Subtendent Then by the Golden rule we find that the multiplying of the middle termes which is nothing else but the squaring of the comprehending sides of the prime Rectangular affords two products equall to the oblongs made of the great Subtendent and his respective Segments the aggregat whereof by equation is the same with the square of the chief Subtendent or Hypotenusa which was to be demonstrated 15. In every totall square the supplements about the partiall and Interior squares are equall the one to the other for by drawing a Diagonall line the great square being divided into two equall Triangles because of their standing on equall bases betwixt two paralells by the ninth Apodictick it is evident that in either of these great Triangles there being two partiall ones equall to the two of the other each to his own by the same Reason of the ninth If from equall things viz. the totall Triangles be taken equall things to wit the two pairs of partiall Triangles equall things must needs remain which are the foresaid supplements whose equality I undertook to prove 16. If a right line cut into two equall parts be increased the square made of the additonall line and one of the Bisegments joyned in one lesse by the Square of the half of the line Bisected is equall to the oblong contained under the prolonged line and the line of Continuation for if annexedly to the longest side of the proposed oblong be described the foresaid Square there will jet out beyond the Quadrat Figure a space or Rectangle which for being powered by the Bisegment and Additionall line will be equall to the neerest supplement and consequently to the other the equality of supplements being proved by the last Apodictick by vertue whereof a Gnomon in the great Square lacking nothing of its whole Area but the space of the square of the Bisected line is apparent to equalize the Parallelogram proposed which was to be demonstrated 17. From hence proceedeth this Sequell that if from any point without a circle two lines cutting it be protracted to the other extremity thereof making two cords the oblongs contained under the totall lines and the excesse of the Subtenses are equall one to another for whether any of the lines passe through the Center or not if the Subtenses be Bisected seeing all lines from the Center fall Perpendicularly upon the Chordall point of Bisection because the two semi-Diameters and Bisegments substerned under equall Angles in two Triangles evince the equality of the third Angle to the third by the fift Apodictick which two Angles being made by the falling of one right line upon another must needs be right by the tenth definition of the first of Euchilde the Bucarnon of Pythagoras demonstrated in my fourteenth Apodictick will by Quadrosubductions of Ambients from one another and their Quadrobiquadrequation● with the Hypotenusa together with other Analogies of equation with the powers of like Rectangular Triangles comprehended within the same circle manifest the equality of long Squares or oblongs Radically meeting in an Exterior point and made of the prolonged Subtenses and the lines of interception betwixt the limb of the circle and the point of concourse quod probandum fuit 18. Now to look back on the eleaventh Apodictick where according to Petiscus