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 ...

CHAP. I. (Book 1)

Definitions Geometrical. (Book 1)

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 A∣rithmetick, and the science of Magni∣tude 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, Geome∣try 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 com∣petent knowledge in Arithmetick.

4 Concerning then this science of Magnitudes, two things are to be con∣sidered. 1 The several heads to which all Magnitudes may be referred. 2 The terms and limits of those Mag∣nitudes.

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, with∣out breadth, or thickness: whether it be, a right line or a crooked, and may be divided into parts in respect oft•• length, but admitteth no other division as the line AB.

[illustration]

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 se∣cond kind of magnitude, to which be∣longeth 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 in∣closing 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.

[illustration]

9 A body or solid is the third kind of magnitude, and hath three dimen∣sions, 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 su∣perficies, 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 con∣tained under one or many limits, un∣der one bound or limit, is comprehen∣ded a Circle, and all other figures un∣der many.

11 A Circle is a figure contained under one round line, which is the cir∣cumference thereof: Thus the round line CBDE is called the circumfe∣rence 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 circumfe∣rence 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 di∣videth 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 Semi∣diameter of the Circle CBDE.

[illustration]

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 Semicir∣cle 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 cir∣cle, 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 se∣veral 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.

19 Right lined parallels are two

[illustration]

right lines equidistant in all places one from another, which being drawn to an infinite length would never meet or concur. As the right lines FHG & CAD in the annexed Diagram.

20 A Circular parallel is a circle drawn within or without another cir∣cle, 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 there∣fore parallel to one another.

21 An angle is the meeting of two lines in any sort, so as they both make not one line: as the two lines AB

[illustration]

and AC incline the one to the other and touch one another in the point A, in which point is made the angle BAC. Where note, that an angle is for the most part described by three letters of which the second or middle letter repre∣senteth the angular point. Thus in the angle BAC the letter A representeth the angular point.

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 croo∣ked line, as the angle GHI.

22 All angles are either right or oblique.

23 A right angle is an angle con∣tained between two right lines, drawn perpendicular to one another. Thus the angle. ABC is a right angle, be∣cause the right line AB is perpendicu∣lar to the right line CD and the contrary.

24 An oblique angle, is an angle contained between to right lines not perpendicular to one another: as the angle CBE or EBD, the one whereof

[illustration]

[viz.] EBD is acute or less then a right angle, and the other [viz.] CBE is ob∣tuse or more then a right, & are the complements of each other to a Se∣micircle.

25 The measure of an angle is the arch of a circle described on the an∣gular point, and intercepted between the two sides of the angle.

[illustration]

Thus in the annex∣ed Dia∣gram, the arch AB is the measure of the an∣gle

AEB, & that the quantity there∣of 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 wan∣teth of a Semicircle; as if the arch BED be 120 degrees, the complement thereof is the arch AB 60 deg.

28 When a right line falling upon two right lines doth make on one and the same side, and two inward angles.

less the two right, the right lines being produced will at length concur in that part in which the two inward angles are less then two right. As in the figure, the right line AB falling upon the right lines CD and EF maketh the angles CGH and GHE together less

[illustration]

then two right angles, if therefore the lines CD and EF be produced from C and E they will at last concur.

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 contain∣ed 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 there∣fore called Triangles, for the better understanding whereof we will here set down some necessary and funda∣mental Propositions of Geometry.