Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

I. By its SIDES.

2. An EQUILATERAL TRIANGLE, is that which hath all its Three Sides equal; as the Figure ABC That is, AB=BC=AC.

B

A

D

3. An IsosCELES TRIANGLE, is that which hath only Two of its Sides equal, as the Figure BDG: That is, BDDG; but the Third Side B G may be either greater or lefs, as Occafion requires.

[blocks in formation]

5. A RIGHT-ANGLED Triangle, is that which hath one Right Angle; that is, when Two of its Sides are Perpendicular to each other, as CA is fuppofed to be to B A. Therefore the Angle at A, is a Right Angle, per Defin. 8. Sect. 1.

[blocks in formation]

Note, The longest Side of every Right-angled Triangle (as BC) is called the Hypotenuse, and the longest of the other Two Sides which include the Right Angle (as B A) is called the Bafs: The Third Side (as CA) is called the Cathetus or Perpendicular.

6. An OBTUSE-ANGLED Triangle, is that which hath one of its Angles Obtufe, and it's called an Amblygonium Triangle. Such is the Third Triangle HKM.

7. An ACUTE-ANGLED TRIANGLE, is that which hath all its Angles Acute, and it's called an Oxygonium Triangle; fuch are the First and Second Triangles ABC, and BDG.

Note, All Triangles that have not a Right Angle, whether they are Acute, or Obtufe, are in general Terms, called Oblique Trian

glesa

[ocr errors]

gles, without any other Diftinction, as before. And the longeft Side of every Oblique Triangle is ufually called the Bafe; the other two are only called Sides or Legs.

8. The Altitude or height of any Plain Triangle, is the Length of a Right Line let fall perpendicular from any of its Angles, upon the Side oppofite to that Angle from whence it falls; and may be either within, or without the Triangle, as Occafion requires, being denoted by the Two prick'd Lines, in the annexed Triangles.

Sect. 4. Of Four-sided Figures.

1. A Square is a plain regular Figure, whofe Area is limited by Four equal Sides all perpendicular one to another.

That is, when AB=BC=CD=DA, and the Angles A, B, C, D are all equal, then it's usually called a Geometrical Square.

2. A Rhombus, or Diamond-like Figure, is that which hath Four equal Sides, but no Right-angle. That is, a Rhombus is a Square mov'd out of its right Pofition, as the annexed Figure.

A

D

B.

3. A Rectangle, or a Right-angled Parallelogram (often called an Oblong, or long Square) is a Fi B gure that hath four Right-angles

and its two oppofite Sides equal, viz. BC HD and BH CD.

[ocr errors]

D

4. A Rhomboides, is an Oblique-angled Parallelogram; that is, it is a Parallelogram moved out of its right Pofition, like the annexed Figure.

5. The Altitude or Height of any Oblique-angled Parallelogram, viz, either of the Rhombus or Rhombaides, is a Right-line let fall perpendicular · from any Angle upon the Side oppofite to that Angle; and may either be within or without the Figure: As the prick'd Lines in the annexed Figure.

PP

6. Every

[blocks in formation]

7. A Right-line, drawn from any Angle in a Four-fided Figure to its oppofite Angle, is called a Diagqual Line, and will divide the Area of the Figure into two Triangles, being denoted by the prick'd Line AC in the last Figure.

8. All Right-lin'd Figures, that have more than four Sides, are call'd Polygons, whether they be regular or irregular.

9. A Regular Polygon is that which hath all its Sides equal, ftanding at equal Angles, and is named according to the Number of its Sides (or Angles.) That is, if it have five equal Sides, it is called a Pentagon; if fix equal Sides, it is call'd a eragon; if seven, 'tis a Heptagon; if eight, 'tis an Daagon, &c.

Note, All Regular Polygons may be infcrib'd in a Circle; that is, their Angular Points, how many foever they have, will all just touch the Circle's Periphery.

10. An Irregular Polygon is that Figure which hath many unequal Sides ftanding at unequal Angles (like unto the annexed Figure, or otherwife); and of fuch Kind of Polygons there are infinite Varieties, but they may all be reduced to regular Figures by drawing Diagonal Lines in them; as fhall be fhew'd farther

on.

These are the most general and ufeful Definitions that concern plain or fuperficial Geometry.

As for those which relate to Solids, I thought it convenient to omit given any Account of them in this Place, because they would rather puzzle and amuse the Learner, than improve him, until he has gain❜d a competent Knowledge in the most useful Theorems concerning Superficies; for then thofe Definitions may be more eafily understood, and will help them to form a clearer Idea of their reSpective Solids, than 'tis poffible to conceive of them before; and therefore I have referv'd thofe Definitions until we come to the Fifth Part.

Sect.

Sect. 5. Of fuch Terms as are generally ufed in Geometry.

Whatsoever is propofed in Geometry will either be a Problem or a beorem.

. Both which Euclid includes in the general Term of Propofition. A Problem is that which proposes fomething to be done, and rèlates more immediately to practical than fpeculative Geometry; That is, it's generally of fuch a Nature, as to be perform'd by fome known or Commonly-receiv'd Rules, without any Regard had to their Inventions or Demonftrations.

A Theorem is when any Commonly-receiv'd Rule, or ony New Propofition is required to be demonftrated, that fo it may from thence forward become a certain Rule, to be rely'd upon in Practice when Occafion requires it. And therefore feveral Rules are often call'd Theorems, by which Operations in Arithmetick, and Conclufions in Geometry, are perform'd.

Note, By Demonstration is underflood the higheft Degree of Proof that human Reafon is capable of attaining to, by a Train of Arguments deduced or drawn from fuch plain Axioms, and other Self-evident Truths, as cannot be denied by any one that confiders them.

A Corollary, or Confectary, is fome Confequent Truth drawn or gain'd from any Demonftration.

A Lemma is the Demonftration of fome Premifes laid down or proposed as preparative to obviate and fhorten the Proof of the The orem under Confideration.

A Scholium is a brief Commentary or Obfervation made upon fome precedent Difcourfe.

N. B. I advise the young Geometer to be very perfect in the Definitions, viz. Not to rest fatisfied with a bare Remembrance of them; but, that he endeavour to gain a clear Idea of Understanding of the Things defined; and for that Reafon I have been fuller in every Definition than is ufual.

And, that he may know from whence most of the following Problems and Theorems contain'd in the Two next Chapters are collected, I have all along cited the Propofition, and Book of Euclid's Elements where they may be found.

As for Inflance; at Problem 1. there is (3. e. 1.) which fhews that it is the Third Propofition in Euclid's Firft Book. The like must be underflood in the Theorems,

[blocks in formation]

CHA P. II.

The First Rudiments, or Leading and Preparatory
Problems, in Plain Geometry.

IN order to perform the following Problems, the young Geometer
ought to be provided with a thin freight Ruler, made either of
Brafs or Box-wood, and two Pair of very good Compaffes, viz. one
Pair called Three-pointed Compafles, being very useful for draw-
ing of Figures or Schemes, either with Black Lead or Ink; and one
Pair of plain Compaffes with very fine Points, to measure and fet off
Distances; alfo he should have a very good Steel Drawing Pen: And
then he may proceed to the Work with this Caution; that he ought to
make himself Mafier of one Problem before he undertakes the next:
That is, he ought to understand the Defign, and, as far as he can, the
Reafon of every Problem, as well as how to do it; and then a little
Practice will render them very easy, they being all grounded upon thefe
following Poftulates.

Poftulates or Petitions.

1. That a Right-line may be drawn from any one given Point

to another.

2. That a Right-line may be produced, encreafed, or made longer from either of its Ends.

3. That upon any given Point (or Center) and with any given Distance (viz. with any Kadius) a Circle may be defcribed.

PROBLEM I.

Two Right-lines being given, to find their Sum and
Difference. (3. e. 1.)

Let the given Lines be

Make the shortest Line, as C B, Radius, and with it defcribe a Cirtle: From its Center C fet off the other Line AC, and join ACB with a Right-line. Then will A B ACCB; and ADA C-GB; as was required.

B

B

[ocr errors]

PRO

« ΠροηγούμενηΣυνέχεια »