THE

ANALY S T;

OR, A

DISCOURSE

Addreffed to an

Infidel MATHEMATICIAN.

WHEREIN

It is examined whether the Object, Principles, and Inferences of the modern Analyfis are more diftinctly conceived, or more evidently deduced, than Religious Myfteries and Points of Faith.

By the Author of The Minute Philofopher.

Firft caft out the beam out of thine own Eye; and then shalt thou fee clearly to caft out the mote out of thy brother's Eye. S. Matt. c. vii. 5.

DUBLIN:

Printed by and for S. FULLER at the Globe in Meath-street, and J. LEATHLY Bookfeller in Dames-ftreet. 1734.

CONTENTS.

1. Mathematicians presumed to be the great masters of reason. Hence

an undue deference to their decisions where they have no right

to decide. This one cause of infidelity.

2. Their principles and methods to be examined with the same freedom

which they assume with regard to the principles and mysteries of

religion. In what sense and how far geometry is to be allowed

an improvement of the mind.

3. Fluxions the great object and employment of the profound geo-

metricians in the present age.
What these fluxions are.

4. Moments or nascent increments of flowing quantities difficult to

conceive. Fluxions of different orders. Second and third

fluxions obscure mysteries.

5. Differences, i.e. increments or decrements infinitely small, used by

foreign mathematicians instead of fluxions or velocities of nascent

and evanescent increments.

6. Differences of various orders, i.e. quantities infinitely less than

quantities infinitely little; and infinitesimal parts of infinitesimals

of infinitesimals, &c. without end or limit.

7. Mysteries in faith unjustly objected against by those who admit

them in science.

8. Modern Analysts supposed by themselves to extend their views

even beyond infinity: deluded by their own species or symbols.

9. Method for finding the fluxion of a rectangle of two indeterminate

quantities, shewed to be illegitimate and false.

10. Implicit deference of mathematical men for the great author of

fluxions. Their earnestness rather to go on fast and far, than to

set out warily and see their way distinctly.

11. Momentums difficult to comprehend. No middle quantity to be

admitted between a finite quantity and nothing, without ad-

mitting infinitesimals.

12. The fluxion of any power of a flowing quantity. Lemma premised

in order to examine the method for finding such fluxion.

13. The rule for the fluxions of powers attained by unfair reasoning.

14. The aforesaid reasoning farther unfolded, and shewed to be illogical.

15. No true conclusion to be justly drawn by direct consequence from

inconsistent suppositions. The same rules of right reason to be

observed, whether men argue in symbols or in words.

16. An hypothesis being destroyed, no consequence of such hypothesis

to be retained.

17. Hard to distinguish between evanescent increments and infinitesimal

differences. Fluxions placed in various lights. The great author,

it seems, not satisfied with his own notions.

18. Quantities infinitely small supposed and rejected by Leibnitz and

his followers. No quantity, according to them, greater or smaller

for the addition or subduction of its infinitesimal.

19. Conclusions to be proved by the principles, and not principles by

the conclusions.

20. The geometrical Analyst considered as a logician; and his dis-

coveries, not in themselves, but as derived from such principles

and by such inferences.

21. A tangent drawn to the parabola according to the calculus differ-

entialis. Truth shewn to be the result of error, and how.

22. By virtue of a twofold mistake Analysts arrive at truth, but not at

science: ignorant how they come at their own conclusions.

23. The conclusion never evident or accurate, in virtue of obscure or

inaccurate premises. Finite quantities might be rejected as well

as infinitesimals.

24. The foregoing doctrine farther illustrated.

25. Sundry observations thereupon.

26. Ordinate found from the area by means of evanescent increments.

27. In the foregoing case, the supposed evanescent increment is really

a finite quantity, destroyed by an equal quantity with an opposite

sign.

28. The foregoing case put generally. Algebraical expressions com-

pared with the geometrical quantities.

29. Correspondent quantities algebraical and geometrical equated. The

analysis shewed not to obtain in infinitesimals, but it must also

obtain in finite quantities.

30. The getting rid of quantities by the received principles, whether of

fluxions or of differences, neither good geometry nor good logic.

Fluxions or velocities, why introduced.

31. Velocities not to be abstracted from time and space: nor their pro-

portions to be investigated or considered exclusively of time and

space.

32. Difficult and obscure points constitute the principles of the modern

Analysis, and are the foundation on which it is built.

33. The rational faculties whether improved by such obscure analytics.

34. By what inconceivable steps finite lines are found proportional to

fluxions. Mathematical infidels strain at a gnat and swallow a

camel.

35. Fluxions of infinitesimals not to be avoided on the received prin-

ciples. Nice abstractions and geometrical metaphysics.

36. Velocities of nascent or evanescent quantities, whether in reality

understood and signified by finite lines and species.

37. Signs or exponents obvious; but fluxions themselves not so.

38. Fluxions, whether the velocities with which infinitesimal differences

are generated.

39. Fluxions of fluxions or second fluxions, whether to be conceived as

velocities of velocities, or rather as velocities of the second

nascent increments.

40. Fluxions considered, sometimes in one sense, sometimes in another;

one while in themselves, another in their exponents: hence con-

fusion and obscurity.

41. Isochronal increments, whether finite or nascent, proportional to

their respective velocities.

42. Time supposed to be divided into moments: increments generated

in those moments: and velocities proportional to those incre-

ments.

43. Fluxions, second, third, fourth, &c., what they are, how obtained,

and how represented. What idea of velocity in a moment of

time and point of space.

44. Fluxions of all orders inconceivable.

45. Signs or exponents confounded with the fluxions.

46. Series of expressions or of notes easily contrived. Whether a series

of mere velocities, or of mere nascent increments corresponding

thereunto, be as easily conceived.

47. Celerities dismissed, and instead thereof ordinates and areas intro-

duced. Analogies and expressions useful in the modern quadra-

tures, may yet be useless for enabling us to conceive fluxions.

No right to apply the rules without knowledge of the principles.

48. Metaphysics of modern Analysts most incomprehensible.

49. Analysts employed about notional shadowy entities. Their logics

as exceptionable as their metaphysics.

50. Occasion of this address. Conclusion. Queries.