may be assigned thereof, sticks so fast that it is impossible without
obstinate striving and labour of the mind to get entirely clear of it.
But then the reluctancy we find in rejecting any opinion can be no
argument of its truth to whoever considers what has been already shown
with regard to the prejudices we entertain concerning the distance,
magnitude, and situation of objects; prejudices so familiar to our minds,
so confirmed and inveterate, as they will hardly give way to the clearest
demonstration.
147. Upon the whole, I think we may fairly conclude that the proper
objects of vision constitute an universal language of the Author of
Nature, whereby we are instructed how to regulate our actions in order to
attain those things that are necessary to the preservation and well-being
of our bodies, as also to avoid whatever may be hurtful and destructive
of them. It is by their information that we are principally guided in all
the transactions and concerns of life. And the manner wherein they
signify and mark unto us the objects which are at a distance is the same
with that of languages and signs of human appointment, which do not
suggest the things signified by any likeness or identity of nature, but
only by an habitual connexion that experience has made us to observe
between them.
148. Suppose one who had always continued blind be told by his guide that
after he has advanced so many steps he shall come to the brink of a
precipice, or be stopped by a wall; must not this to him seem very
admirable and surprizing? He cannot conceive how it is possible for
mortals to frame such predictions as these, which to him would seem as
strange and unaccountable as prophesy doth to others. Even they who are
blessed with the visive faculty may (though familiarity make it less
observed) find therein sufficient cause of admiration. The wonderful art
and contrivance wherewith it is adjusted to those ends and purposes for
which it was apparently designed, the vast extent, number, and variety of
objects that are at once with so much ease and quickness and pleasure
suggested by it: all these afford subject for much and pleasing
speculation, and may, if anything, give us some glimmering analogous
prenotion of things which are placed beyond the certain discovery and
comprehension of our present state.
149. I do not design to trouble myself with drawing corollaries from the
doctrine I have hitherto laid down. If it bears the test others may, so
far as they shall think convenient, employ their thoughts in extending it
farther, and applying it to whatever purposes it may be subservient to:
only, I cannot forbear making some inquiry concerning the object of
geometry, which the subject we have been upon doth naturally lead one to.
We have shown there is no such idea as that of extension in abstract, and
that there are two kinds of sensible extension and figures which are
entirely distinct and heterogeneous from each other. Now, it is natural
to inquire which of these is the object of geometry.
150. Some things there are which at first sight incline one to think
geometry conversant about visible extension. The constant use of the
eyes, both in the practical and speculative parts of that science, doth
very much induce us thereto. It would, without doubt, seem odd to a
mathematician to go about to convince him the diagrams he saw upon paper
were not the figures, or even the likeness of the figures, which make the
subject of the demonstration. The contrary being held an unquestionable
truth, not only by mathematicians, but also by those who apply themselves
more particularly to the study of logic; I mean, who consider the nature
of science, certainty, and demonstration: it being by them assigned as
one reason of the extraordinary clearness and evidence of geometry that
in this science the reasonings are free from those inconveniences which
attend the use of arbitrary signs, the very ideas themselves being copied
out and exposed to view upon paper. But, by the bye, how well this agrees
with what they likewise assert of abstract ideas being the object of
geometrical demonstration I leave to be considered.
151. To come to a resolution in this point we need only observe what hath
been said in sect. 59, 60, 61, where it is shown that visible extensions
in themselves are little regarded, and have no settled determinable
greatness, and that men measure altogether, by the application of
tangible extension to tangible extension. All which makes it evident that
visible extension and figures are not the object of geometry.
152. It is therefore plain that visible figure are of the same use in
geometry that words are: and the one may as well be accounted the object
of that science as the other, neither of them being otherwise concerned
therein than as they represent or suggest to the mind the particular
tangible figures connected with them. There is indeed this difference
between the signification of tangible figures by visible figures, and of
ideas by words: that whereas the latter is variable and uncertain,
depending altogether on the arbitrary appointment of men, the former is
fixed and immutably the same in all times and places. A visible square,
for instance, suggests to the mind the same tangible figure in Europe
that it doth in America. Hence it is that the voice of the Author of'
Nature which speaks to our eyes, is not liable to that misinterpretation
and ambiguity that languages of human contrivance are unavoidably subject
to.
153. Though what has been said may suffice to show what ought to be
determined with relation to the object of geometry, I shall nevertheless,
for the fuller illustration thereof, consider the case of an
intelligence, or unbodied spirit, which is supposed to see perfectly
well, i.e. to have a clear perception of the proper and immediate objects
of sight, but to have no sense of touch. Whether there be any such being
in Nature or no is beside my purpose to inquire. It sufficeth that the
supposition contains no contradiction in it. Let us now examine what
proficiency such a one may be able to make in geometry. Which speculation
will lead us more clearly to see whether the ideas of sight can possibly
be the object of that science.
154. FIRST, then, it is certain the aforesaid intelligence could have no
idea of a solid, or quantity of three dimensions, which followeth from
its not having any idea of distance. We indeed are prone to think that we
have by sight the ideas of space and solids, which ariseth from our
imagining that we do, strictly speaking, see distance and some parts of
an object at a greater distance than others; which hath been demonstrated
to be the effect of the experience we have had, what ideas of touch are
connected with such and such ideas attending vision: but the intelligence
here spoken of is supposed to have no experience of touch. He would not,
therefore, judge as we do, nor have any idea of distance, outness, or
profundity, nor consequently of space or body, either immediately or by
suggestion. Whence it is plain he can have no notion of those parts of
geometry which relate to the mensuration of solids and their convex or
concave surfaces, and contemplate the properties of lines generated by
the section of a solid. The conceiving of any part whereof is beyond the
reach of his faculties.
155. Farther, he cannot comprehend the manner wherein geometers describe
a right line or circle; the rule and compass with their use being things
of which it is impossible he should have any notion: nor is it an easier
matter for him to conceive the placing of one plane or angle on another,
in order to prove their equality: since that supposeth some idea of
distance or external space. All which makes it evident our pure
intelligence could never attain to know so much as the first elements of
plane geometry. And perhaps upon a nice inquiry it will be found he
cannot even have an idea of plane figures any more than he can of solids;
since some idea of distance is necessary to form the idea of a
geometrical plane, as will appear to whoever shall reflect a little on
it.
156. All that is properly perceived by the visive faculty amounts to no
more than colours, with their variations and different proportions of
light and shade. But the perpetual mutability and fleetingness of those
immediate objects of sight render them incapable of being managed after
the manner of geometrical figures; nor is it in any degree useful that
they should. It is true there are divers of them perceived at once, and
more of some and less of others: but accurately to compute their
magnitude and assign precise determinate proportions between things so
variable and inconstant, if we suppose it possible to be done, must yet
be a very trifling and insignificant labour.
157. I must confess men are tempted to think that flat or plane figures
are immediate objects of sight, though they acknowledge solids are not.
And this opinion is grounded on what is observed in painting, wherein (it
seems) the ideas immediately imprinted on the mind are only of planes
variously coloured, which by a sudden act of the judgment are changed
into solids. But with a little attention we shall find the planes here
mentioned as the immediate objects of sight are not visible but tangible
planes. For when we say that pictures are planes, we mean thereby that
they appear to the touch smooth and uniform. But then this smoothness and
uniformity, or, in other words, this planeness of the picture, is not
perceived immediately by vision: for it appeareth to the eye various and
multiform.
158. From all which we may conclude that planes are no more the immediate
object of sight than solids. What we strictly see are not solids, nor yet
planes variously coloured: they are only diversity of colours. And some
of these suggest to the mind solids, and other plane figures, just as
they have been experienced to be connected with the one or the other: so
that we see planes in the same way that we see solids, both being equally
suggested by the immediate objects of sight, which accordingly are
themselves denominated planes and solids. But though they are called by
the same names with the things marked by them, they are nevertheless of a
nature entirely different, as hath been demonstrated.
159. What hath been said is, if I mistake not, sufficient to decide the
question we proposed to examine, concerning the ability of a pure spirit,
such as we have described, to know GEOMETRY. It is, indeed, no easy
matter for us to enter precisely into the thoughts of such an
intelligence, because we cannot without great pains cleverly separate and
disentangle in our thoughts the proper objects of sight from those of
touch which are connected with them. This, indeed, in a complete degree
seems scarce possible to be performed: which will not seem strange to us
if we consider how hard it is for anyone to hear the words of his native
language pronounced in his ears without understanding them. Though he
endeavour to disunite the meaning from the sound, it will nevertheless
intrude into his thoughts, and he shall find it extreme difficult, if not
impossible, to put himself exactly in the posture of a foreigner that
never learned the language, so as to be affected barely with the sounds
themselves, and not perceive the signification annexed to them.
160. By this time, I suppose, it is clear that neither abstract nor
visible extension makes the object of geometry; the not discerning of
which may perhaps have created some difficulty and useless labour in
mathematics. Sure I am, that somewhat relating thereto has occurred to my
thoughts, which, though after the most anxious and repeated examination I
am forced to think it true, doth, nevertheless, seem so far out of the
common road of geometry, that I know not whether it may not be thought
presumption, if I should make it public in an age, wherein that science
hath received such mighty improvements by new methods; great part whereof,
as well as of the ancient discoveries, may perhaps lose their reputation,
and much of that ardour with which men study the abstruse and fine
geometry be abated, if what to me, and those few to whom I have imparted
it, seems evidently true, should really prove to be so.