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 those in Spherical being deduced from the Lord Nepeirs Catholick or Universal proposition By J. NEWTON M. A. LONDON Printed for George Hurlock at Magnus Church-corner and Thomas Pierrepont at the Sun in Paul's Church-yard 1659. TO THE READER MAthematical Sciences in their full latitude doe comprehend all Arts and Sciences whatsoever but in a more strict sense those onely are esteemed such which doe belong to the Doctrine of Quantity amongst these this of Triangles is not of least concernment of which though many have written and that excellently yet still there may be something added at least by way of explanation if not some matter absolutely new What we have here puplished is intended for beginners onely in which we have endeavored after brevity perspicuity both in the construction of the Canon of natural sines Tangents and Secants first then the Axioms Problems for the solution of all Triangles both plain Spherical either by those natural numbers or by the Logarith of them in the solution of plain Triangles by logarithmetical numbers sometimes the logarithmes of a Decimal fraction is required for the finding whereof having given no direction in our Preface to those Tables we have though somewhat out of place inserted it her A fraction whether Vulgar or Decimal being given to find the logarithme thereof That the way method here propounded may be the better conceived the proper charecteristique to the logar of any integer or whole number must be considered and the characteristique of the log arithmes of all numbers under 10 is 0 of all numbers hetween 10 and 100 is 1 between 100 1000 is 2 so forward nor will there as I couceive be any inconvenience if 10 be the characteristique of the logarithme of any digit 11 the characteristique of all numbers between 10 100 12 the characteristique of all numbers between 100 1000 so forward yet if this shall be supposed the characteristique of a number that is but one place beneath unitie shall be 9 if it be two places beneath unitie it shall be 8 c. Thus the Log of 5 will be 10.698970 The Log of 5 tenths will be 9.698970 The Log of 05 hundreds will be 8.698970 And upon this ground to find the Log of a vulgar fraction you must subtract the logarithme of the denominator from the logarithme of the numerator and what remaineth shall be the logarithme of the vulgar fraction given Example let 6 12 be the vulgar fraction given logar. 6 the Numeâ—ator is 0.778151 logar. 12 the Denominator is 1.079181 There difference 9.698970 is the log of 6 12 the fraction propounded And thus the logarithme of any fraction or the decimal fraction answering to any logarithme may be as easily found as by any directions hitherto given and in their use are much more ready as will be manifest by the Example following If 6 tenths of an ounce of Gold cost .95 hundreds of a pound sterling what shall .07 hundreds of an ounce of the same gold cost Answer is 1108. Logar. 6 tenths is 9.778151 logar. 95 hundreds is 9.977724 logar. 07 hundreds is 8.845098 1.8822822 Adswer 1108 9â—044671 To resolve this question you must adde the Logarithmes of the second third terms together from their sum subtract the first what remaineth will be the logar. of the fourth proportional required the characteristique of which fourth proportional doth by inspection shew whether it be the logarithme of a fraction or not there is but one difficulty in this way of working which is to substract a greater number from a less in which if you will but borrow from a supposed figure as we doe from a real and set down the difference that difficultie is also avâ—ided as in the first Example Though I can subtract 1 out of 1 yet I cannot subtract 8 out of 5 but borrowing one from the next figure I can subtract 8 out of 15 and there will remain 7 and so proceeding till you come the characteristique the remainer will be found in the ordinary way but the characteristigue of the upper number being a cypher and that of the lower an unite I cannot subtract 1 from 0 but supposing an unite to be beyond the cypher I can subtract 1 from 10 and there will remainer 9 the like is to be done in other cases which I must leave to the consideration of the Ingenious Reader here being no room for to explaim my self JOHN NEWTON An explication of the Symbols Equal to More − Less × Drawn into ∷ Proportionality Z The Sum X The Difference s Sine cs Co-sine t Tangent ct Co-tangent R. ang. Right angle 2 R. ang. Two right angles q. Square CHAP. I. Definitions Geometrical OF things Mathematical there are two principal kinds Number and Magnitude and each of these hath his proper science 2 The science of Number is Arithmetick and the science of Magnitude is commonly called Geometry but may more properly be termed Magethelogia as comprehending all Magnitudes whatsoever whereas Geometry by the very Etymologie of the word doth seeme to confine this science to Land measuring onely 3 Of this Magethelogia Geometry or science of Magnitudes we will set down such grounds and principles as are necessary to be known for the better understanding of that which followeth presuming that the Reader hereof hath already gotten some competent knowledge in Arithmetick 4 Concerning then this science of Magnitudes two things are to be considered 1 The several heads to which all Magnitudes may be referred 2 The terms and limits of those Magnitudes 5 All magnitudes are either lines planes or solids and doe participate of one or more of these dimensions length breadth and thickness 6 A line is a supposed length or a thing extending it self in length without breadth or thickness whether it be a right line or a crooked and may be divided into parts in respect oftâ— length but admitteth no other division as the line AB 7 The ends or limits of a line are points as having his beginning from a point and ending in a point and therefore a point hath neither part nor quantity As the points at A B are the ends of the afore-saidline AB and no parts thereof 8 A plain or superficies is the second kind of magnitude to which belongeth two dimensions length and breadth but not thickness As the end limits or bounds of a line are points confining the line So lines are the limits bounds and ends inclosing a superficies as in the figure A BCD the plain or superficies thereof is inclosed with the four lines AB BD DC CA which are the extreams or limits thereof 9 A body or
solid is the third kind of magnitude and hath three dimensions length breadth and thickness And as a point is the limit or term of a line and a line the limit or term of a superficies So likewise a superficies is the end and limit of a body or solid and representeth to the eye the shape or figure thereof 10 A Figure is that which is contained under one or many limits under one bound or limit is comprehended a Circle and all other figures under many 11 A Circle is a figure contained under one round line which is the circumference thereof Thus the round line CBDE is called the circumference of that Circle 12 The center of a Circle is the point which is in the middest thereof from which point all right lines drawn to the circumference are equal to one another As in the following figure the lines AB AC and AD are equal 13 The Diameter of a Circle is a right line drawn through the center thereof and ending at the circumference on the other side dividing the circle into two equal parts As the lines CAD and BAE are either of them the Diameter of the Circle CBDE because that either of them doth pass through the Center A and divideth the Circle into two equal parts 14 The Semidiameter of a circle is halfe the Diameter and is contained between the center and one side of the Circle As the lines AB AC AD and AE are either of them the Semidiameter of the Circle CBDE 15 A Semicircle is the one half of a circle drawn upon his Diameter is contained by the half circumference and the Diameter As the Semicircle CBD is half the Circle CBDE and drawn upon the Diameter CAD 16 A quadrant is the fourth part of a Circle and is contained between the Semidiameter of the circle and a line drawn perpendicular unto the Diameter of the same circle from the center thereof dividing the Semicircle into two equal parts of the which parts the one is the quadrant or fourth part of the circle Thus from the center A the perpendicular AB being raised perpendicularly upon the Diameter CAD divideth the Semîcircle CBD into the two equal parts CFB FGD each of which is a Quadrant or fourth part of the circle CBDE 17 A Segment or portion of a circle is a figure contained under a right line and a part of the circumference of a circle either greater or lesser then a Semicircle As in the former figure FBGH is a Segment or part of the circle CBDE contained under the right line FHG less then the Semicircle CBD And by the application of the several lines or terms of a superficies one to another are made parallels angles and many sided figures 18 A parallel line is a line drawn by the side of another line in such sort that they may be equi-distant in all places and of such there are two sorts the right lined parallel and the circular parallel 20 A Circular parallel is a circle drawn within or without another circle upon the same center as you may plainly see by the two circles CFGD and ABHE which are both drawn upon the same center K are therefore parallel to one another If the lines which containeth the angle be right lines it is called a right lined angle As the angle BAC A crooked line angle is that which is contained of crooked lines as the angle DEF And a mixt angle is that which is contained both of a right and a crooked line as the angle GHI 22 All angles are either right or oblique 23 A right angle is an angle contained between two right lines drawn perpendicular to one another Thus the angle ABC is a right angle because the right line AB is perpendicular to the right line CD and the contrary 25 The measure of an angle is the arch of a circle described on the angular point and intercepted between the two sides of the angle Thus in the annexed Diagram the arch AB is the measure of the angle AEB that the quantity thereof may be the better known 26 Every circle is supposed to be divided into 360 parts or deg. every degr. into 60 min. or 100 parts c. Therefore a Semicircle as the arch ACD is 180 deg. a quadrant or fourth part of a Circle as the arch ABC is 90 deg. 27 Complements of arches are either in reference to a quadrant or a Semicircle the complement of an arch or angle to a quadrant is so much as the arch given wanteth of a quadrant or 90 deg. as if the arch AB be 60 deg. the complement thereof to a quadrant is the arch BC 30. In like manner the complement of an arch or angle to a Semicircle is so much as the arch or angle given wanteth of a Semicircle as if the arch BED be 120 degrees the complement thereof is the arch AB 60 deg. 29 Many sided figures are such as are made of three four or more lines though for distinction sake those onely are so called which are contained under five lines or terms at the leas In this Treatise we have to do with such onely as are contained under three lines or sides these are therefore called Triangles for the better understanding whereof we will here set down some necessary and fundamental Propositions of Geometry CHAP. II. Propositions Geometrical 1. Prop. If two sides of one Triangle be equal to two sides of another and the angle comprehended by the equal sides equal the third side or base of the one shall be equal to the base of the other and the remaining angles of the one equal to the remaining angles of the other Demonst. In the Triangles CBH FED make CB FE and BH ED and ang. CB FED then shall CH FD For if CG FD the angle CBG FED but CBH FED by construction therefore CH FD ang. C F and the angle H D as was to be proved 2 Prop. If a Triangle have two equal sides the angles at the base are also equal to one another and the contrary Demonst. In the Triangle ABC let AB AC and let AB be extended to D and AE AD and draw the lines BE and DC now then because AD AE and AB AC and the angle A common to the Triangle ADC and ABE the base BE DC the angle D E and ABE ang. ACD by the former prop. and therefore ang. DCB EBC and being taken from the equal angles ABE ACD there shall remain the angle ABC ACB as was to be proved 3 Prop. If two right lines doe cut through one another the angles opposite to one another are equal Demonst. Ang. ACB BCD 2 right ang. 24 of the first ang. ACB ACE 2 right and ang. ABC common therefore ang. BCD ACE as was to be proved 4 If the side of a Triangle be extended the outward angle shall be greater then any one of the inward opposite angles 5 Prop. In all Triangles the greatest sides subtend the greatest
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 ciâ—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