The foregoing short description was deemed necessary, but the uses of the Instruments must be deferred, until the learner has acquired sufficient skill in Geometry, to understand them.

OF GEOMETRY, CONSIDERED AS THE SCIENCE OF DEMONSTRATION.

As the reader is supposed to be unacquainted with logic, it will be proper in this place to introduce a few particulars taken from that art, which may serve as an introduction.

1. The mind becomes conscious of the existence of external objects by the impressions it receives from them. There are five inlets or channels, called the organs of sense, by which the mind receives all its original information; namely, the eye, the ear, the nose, the palate, and the touch: hence seeing, hearing, smelling, tasting, and feeling, are called the five senses. This great source of knowledge, comprehending all the notices conveyed to the mind by impulses made by external objects on the organs of sense, is called SENSATION.

2. PERCEPTION is that whereby the mind becomes conscious of an impression; thus, when I feel cold, I hear thunder, I see light, &c. and am conscious of these effects on my mind, this consciousness is called perception.

3. AN IDEA results from perception; it is the representation or impression of the thing perceived on the mind, and which it has the power of renewing at pleasure.

.

4. The power which the mind possesses of retaining its ideas, and renewing the perception of them, is called MEMORY; and the act of calling them up, examining, and reviewing them, is called REFLECTION.

5. In addition to the numerous class of ideas derived by sensation wholly from without, the mind acquires others by reflection; thus by turning our thoughts inward, and observing what passes in our own minds, we gain the ideas of hope, fear, love, thought, reason, will, &c. The ideas derived by means of sensation are called SENSIBLE IDEAS, and those obtained by reflection, INTELLECTUAL IDEAS.

6. From these two sources alone (viz. sensation and reflection) the mind is furnished with ample store of materials for its future operations; sensation supplies it with the original

stock derived from without, and reflection increases that stock, deriving other ideas by means of it from within..

7. A SIMPLE IDEA is that which cannot be divided into two or more ideas; thus the ideas of green, red, hard, soft, sweet, &c. are simple.

8. A COMPLEX IDEA is that which arises from joining two or more simple ideas together; thus the ideas of beer, wine, falsehood, a house, a square, are complex, being each made up of the ideas of the several ingredients or particulars which compose it, together with that of their manner of combination.

9. In receiving its impressions, the mind is wholly passive; it cannot create one new simple idea: those from without obtrude themselves on it by means of the senses, and those from within, which arise, from the mind's contemplating the impressions it has already received, are equally spontaneous and (with respect to the mind) involuntary. But although the mind cannot create one original simple impression, yet when it is stored with a number of simple ideas, it possesses a wonderful power over them: it can combine several simple ideas together, so as to form a complex one, and vary the combinations at pleasure; it can compare its ideas, and readily determine in what particulars they agree, and in what they disagree. Having combined several simple ideas so as to form a complex one, the mind can again separate or resolve this complex idea into its component simple ones: this it can do both completely, and in part; it can retain just as many of the simple ideas in composition (out of the number which forms the entire complex one) as it chooses, and reject the rest; and if to this arbitrary combination a name be given, whenever we hear that name pronounced, the idea compounded of the whole of the parts prescribed, and no more, occurs immediately to the mind.

10. From the comparison of ideas arises what is called RELATION; and among other relations that which in mathematics is called RATIO, being a relation arising from the comparison of quantities in respect of their magnitude only.

11. In comparing several complex ideas together, we find, that although they differ with respect to some of the simple ideas of which they are compounded, yet they agree in some general character: thus, a triangle and a square differ with respect to

their form, the number of their sides, and the number and magnitude of their angles; but they agree in one general character, they are both figures. A lion and a sheep differ widely from each other in many particulars; but in their general character they agree, viz. they are both animals.

12. This most important power of the mind over its complex ideas is called ABSTRACTION, and the general idea produced by its operation is called an ABSTRACT IDEA.

13. An abstract idea then comprehends in one general class, not only all the simple ideas, but all the complex ones from which it is abstracted: thus the idea of beast is a complex idea, and includes the ideas of lion, horse, bear, wolf, rabbit, &c. the idea of animal is likewise complex, including those of man, beast, bird, fish, insect, &c.

14. Hence an abstract or general idea is merely a creature of the mind, and can have no existing pattern or architype: we can form in the mind the abstract idea of a triangle, viz. one that shall include the ideas of all particular triangles; but we cannot describe on paper any figure capable of representing a triangle in general, viz. all the varieties of triangles that can be made.

15. Hence also whatever is true of an abstract idea is likewise true of every particular complex or simple idea included under it; thus, if it be proved generally that two sides of a triangle are together greater than the third, it follows that the same thing is thereby proved, and must be true of each and every individual triangle: in like manner whatever is proved of plane rectilineal figures in general, will necessarily be true (not only of every kind, but) of every particular rectilineal figure that can be made; thus, since it follows from prop. 32. book 1. of Euclid, that all the interior angles (taken together) of every rectilineal figure are equal to twice as many right angles, wanting four, as the figure has sides, the same thing must be true of each particular kind of such figure; as of squares, triangles, trapeziums, polygons, &c. and likewise of every particular figure included under those kinds.

16. Upon an examination of our ideas of the objects that surround us, we shall find that several of them resemble each other, except in one, two, or perhaps more circumstances ;

now if we leave out from our consideration the particulars in which they disagree, and retain those only in which they agree, we shall obtain the abstract idea of a SPECIES, which, as it is supposed to arise from the lowest possible degree of abstraction, is called THE INFERIOR SPECIES; and the individuals which compose it, being supposed capable of no subordi nate arrangement, are called PARTICULARS. If this idea of species be compared with our ideas of other species, we shall in like manner perceive that they disagree in some of their circumstances only; wherefore by leaving these out as before, we shall obtain the idea of a species superior to the former, viz. which includes the former, and one, two, or more others. In like manner by continual abstraction we pass through the successive gradations of species, until at length we arrive at a point where no further abstraction is possible: the ultimate idea thus obtained, as including the ideas of all the several species, is called a

GENUS.

17. Thus by successive acts of abstraction, a guinea is gold, metal, substance, being; a herring is fish, animal, substance, being; Tray is greyhound, dog, beast, animal, substance, being; an oak is tree, vegetable, substance, being; James is scholar, man, animal, substance, being, &c. In the examples here proposed it may be observed, that substance is common to them all; the idea of substance includes therefore those of metal, animal, and vegetable, and consequently the subordinate ideas of guinea, herring, Tray, oak, and James. Substance then is to be considered as the PROXIMATE GENUS of these, including them all; BEING is the HIGHEST or SUPERIOR GENUS, and implies merely existence.

18. As a general knowledge of the operations of the mind in compounding, comparing, and abstracting its ideas, is necessary to those who would fully understand the plan and scope of Euclid, so it will be equally profitable to shew, in as plain a manner as possible, how our abstract and other complex ideas are unfolded, so as to make them intelligible by words (expressed either by the voice or writing) to others.

19. And first, simple ideas are expressed by words arbitrarily assumed as their representatives; so that whenever any word is read or pronounced, the idea it stands for immediately occurs to the mind of the reader or hearer: but should it happen in any

instance otherwise, the object which produces the idea must be presented to him, and he must be informed that such a word is the sign of that idea; or should the idea have two or three different words to express it, these should all be pronounced, and probably the idea will occur to him from one of the:: there is no other method of communicating a simple idea from one mind to another. I point a person to the object, I tell him its name, and immediately his mind associates the latter with the idea of the former, making the name the constant representative of the idea.

20. But although simple ideas cannot be conveyed to the mind by any verbal description, the case is different with respect to complex ideas; these may be communicated with great facility for since a complex idea is composed of several simple ones, if the names of the latter be pronounced, together with their mode of connection, the complex idea will immediately occur to the hearer; provided his mind be previously furnished with its component simple ideas, together with a knowledge of the names or signs by which they are expressed.

21. It has been shewn, that if the difference between individuals, agreeing in their general and most remarkable properties and circumstances, (and which is called their NUMERAL DIFFERENCE,) be rejected, we obtain the abstract idea of a species; if the difference between this species and another species (called the SPECIFIC DIFFERENCE) be rejected, we get the idea of a species, which includes and is superior to the former; and if in like manner we contínually drop the successive specific differences, we shall at length arrive at the genus, or summit of our research.

22. Hence an easy method presents itself of unfolding a complex idea, or of communicating our complex ideas to other persons by means of definitions, namely by following a contrary order: we name the genus or kind, to this name we join that of the specific difference, and both together will convey to the mind of the hearer the complex idea we mean to describe. Again, if we consider this species as a genus, and join to it the next lower specific difference, the result will give a precise- idea of the next inferior species; proceeding in this manner through all the successive ranks of species to the lowest, to which joining the numeral difference, we at length obtain the idea of a particular