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end_n arch_n line_n right_a 1,244 5 9.1480 5 false
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ID Title Author Corrected Date of Publication (TCP Date of Publication) STC Words Pages
A52262 Geometrical trigonometry, or, The explanation of such geometrical problems as are most useful & necessary, either for the construction of the canons of triangles, or for the solution of them together with the proportions themselves suteable unto every case both in plain and spherical triangles ... / by J. Newton ... Newton, John, 1622-1678. 1659 (1659) Wing N1059; ESTC R17311 30,729 123

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BCq by the 19 hereof 22 If a plaine Triangle be inscrihed in a circle the angles opposite to the circumference are half as much as that part of the circumference which is opposite to the angles Demonst. In the Triangle EBD ang. EDB EBD by the second hereof and ang. AEB equal to both by the 9th hereof the arch AB is the measure of the angle AEB by the 25th of the first therefore the arch AB is the double measure of the angle ADB as was to be proved 1 Consectary If the side of a plaine Triangle inscribed in a Circle be the Diameter the angle opposite to that side is a right angle As the angle ABD opposite to the diameter AD 2 Consectary If divers right lined Triangles be inscribed in the same segment of a circle upon one base the angles in the circumference are equal As the Triang. ABD ACD being inscribed in the same segment of the circle ABCD and upon the same base AD have their angles at B and D falling in the circumference equal 23 If a quadrilateral figure be inscribed in a Circle the angles thereof which are opposite to one another are together equal to two right angles Demonst. Ang. CDB CAB BDA BCA by the last aforegoing therefore ang. CDA BCA BAC and ABC BAC BCA 2 R. ang. by the 9th hereof therefore ang. ABC ADC 2 R. ang. as was to be proved 24 If in a quadrilateral figure inscribed in a circle there be drawn two Diagonal lines the rectangle under the Diagonals is equal to the two rectangles under the opposite sides Demonst. Let ang. DAE CAB by construction then shall ang. DAC EAB and ang. ACD ABE because the arch A D is the double measure to them both and therefore the triangles ADC AEB are like Again ang. ADB ACB because the arch AB is the double measure to them both and ang. DAE CAB by construction the Triang. AED and ABC like therefore AC CB ∷ AD DE And AC CD ∷ AB BE. Therefore AC × DE CB × AD And also AC × BE CD × AB And AC × DE AC × BE AC × DB. Therefore AC × DB CB × AD CD × AB as was to be proved CHAP. III. Of the Construction of the Canon of Triangles THat the Proportions which the parts of a Triangle have one to another may be certain the arches of circles by which the angles of all Triangles and of Spherical Triangles the sides are also measured must be first reduced into right lines by defining the quantity of right lines as they are applyed to the arches of a circle 2 Right lines are applyed to the arches of a circle three wayes viz. either as they are drawn within the circle without the circle or as they are drawn through it 3 Right lines within the circle are Chords and sines 4 A Chord or subtense is a right line inscribed in a ci●cle dividing the whole circle into two segments and in like manner subtending both the segments as the right line CK divideth the circle GEDK into the two segments CEGK and CDK and subtendeth both the segments that is the right line CK is the chord of the arch CGK and also the chord of the arch CDK 5 A Sine is a right line in a semicircle falling perpendicular from the term of an arch 6 A Sine is either right or versed 7 A right Sine is a right line in a Semicircle which from the term of an arch is perpendicular to the diameter dividing the Semicircle into two segments and in like manner referred to both Thus the right line CA is the sine of the arch CD less then a quadrant and also the sine of the arch CEG greater then a quadrant and hence instead of the obtuse angle GBC we take the acute angle CBA the complement thereof to a Semicircle and so our Canon of Triangles doth never exceed 90 deg. 8 A right sine is either Sinus totus that is the Radius or whole Sine as the right line EB or Sinus simpliciter the first sine or a sine less then Radius as AC or AB the one whereof is alwayes the complement of the other to 90 degrees we usually call them sine and co-sine 10 Right lines without the Circle whose quantity we are to define are such as touch the circle and are called Tangents 11 A Tangent is a right line which touching the circle without is perpendicular from the end of the diameter to the Radius continued through the term of that arch of which it is the Tangent Thus the right line FD is the Tangent of the arch CD 12 Right lines drawn through the circle whose quantity we are to define are such as cut the circle and are called Secants 13 The Secant of an arch is a right line drawn through the term of an arch to the Tangent line of the same arch and thus the right line BF is the Secant of the arch CD as also of the arch CEG the complement thereof to a Semicircle 14 A Canon of Triangles then is that which conteineth the Sines Tangents and Secants of all degrees parts of degrees in a quadrant according to a certain diameter or measure of a circle assumed The construction whereof followeth and first of the Sines 15 The right Sines as they are to be considered in order to their construction are either Primary or Secondary 16 The Primary Sines are those by which the rest are found And thus the Radius or whole sine is the first primary sine and is equal to the side of a six-angled figure inscribed in a Circle Consectary The Radius of a circle being given the sine of 30 deg. is also given for by this proposition the Radius of a circle is the subtense of 60 deg. and the half thereof is the sine of 30 and therefore the Radius AB or BC being 1000.0000 the sine of 30 deg. is 500.0000 17 The other primary sines are the sines of 60.18 and 12 deg. being the half of the subtenses of 120. 36 24 degr. and may be found by the problems following 18 The right sine of an arch the right sine of its complement are in power equal to Radius Demonst. In the first diagram of this chapter AC is the sine of CD and AB the sine of CE the complement thereof which with the Radius BC make the right-angled Triangle ABC therefore ABq ACq BCq by the 19 of the second as was to be proved And hence the sine of 60 deg. may thus be found let the sine of 30 deg. AC be 500.0000 the square whereof 250.00000 being subtracted from the square of BC Radius the remainer is 750.00000 the square of AB whose square root is 8660254 the sine of 60 deg. 19 The subtense of 36 deg. is the side of a Dec-angle inscribed in a circle or the greater segment of a Hexagon divided into extream and meane proportion Corsectary