The Method of Fluxions

3. The Method of Fluxions is the general Key that modern Mathematicians use to unlock the secrets of Geometry, and consequently of Nature.

• Lines are supposed to be generated[1] by the motion of Points
• Planes by the motion of Lines
• Solids by the motion of Planes

Quantities generated in equal times are greater or lesser, according to the greater or lesser Velocity, wherewith they increase and are generated, a Method hath been found to determine Quantities from the Velocities of their generating Motions.

• Such Velocities are called Fluxions.
• The Quantities generated are called flowing Quantities.

These Fluxions are said to be nearly as the Increments of the flowing Quantities, generated in the least equal Particles of time.

• These are accurately in the first Proportion of the nascent, or in the last of the evanescent, Increments.

Sometimes, instead of Velocities, the momentaneous Increments or Decrements of undetermined flowing Quantities are considered, under the Appellation of Moments.

4. Moments are not finite Particles.

Finite Particles are Quantities generated from Moments, which last are only the nascent Principles of finite Quantities.

It is said, that the minutest Errors are not to be neglected in Mathematics: that the Fluxions are Celerities, not proportional to the finite Increments though ever so small; but only to the Moments or nascent Increments, whereof the Proportion alone, and not the Magnitude, is considered.

The Fluxions of Fluxions are called second Fluxions.

The Fluxions of these second Fluxions are called third Fluxions: and soon, fourth, fifth, sixth, &c. ad infinitum.

Our Sense is strained and puzzled with the perception of Objects extremely minute.

Our Imagination, which Faculty derives from Sense, is very much strained and puzzled to frame clear Ideas of the least Particles of time, or the least Increments generated therein.

Our Imagination is much more so to comprehend the Moments, or those Increments of the flowing Quantities in statu nascenti, in their very first origin or beginning to exist, before they become finite Particles.

It is still more difficult, to conceive the abstracted Velocities of such nascent imperfect Entities.

But the Velocities of the Velocities, the second, third, fourth and fifth Velocities, &c. exceed all Human Underftanding.

The further the Mind analyses these fugitive Ideas, the more it is lost and bewildered.

The Objects, at first fleeting and minute, soon vanishing out of sight.

Certainly in any Sense a second or third Fluxion seems an obscure Mystery.

The incipient Celerity of an incipient Celerity, the nascent Augment of a nascent Augment, i. e. of a thing which hath no Magnitude:

The clear Conception of it will be found impossible.

If a second Fluxion be inconceivable, what are we to think of third, fourth, fifth Fluxions, and so onward without end?

5. The foreign Mathematicians are said to proceed in a way less accurate and geometrical, yet more intelligible.

Instead of flowing Quantities and their Fluxions, they consider the variable finite Quantities, as increasing or diminishing by the continual Addition or Subduction of infinitely small Quantities.

Instead of the Velocities wherewith Increments are generated, they consider the Increments or Decrements themselves. They call these Differences, and which are supposed to be infinitely small.

The Difference of a Line is an infinitely little Line.

The Difference of a Plane is an infinitely little Plane.

They suppose finite Quantities to consist of Parts infinitely little, and Curves to be Polygones, whereof the Sides are infinitely little, which by the Angles they make one with another determine the Curvity of the Line.

To conceive a Quantity infinitely small, that is, infinitely less than any sensible or imaginable Quantity, or than any the least finite Magnitude, is, I confess, above my Capacity.

But to conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply’d infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever; and will be allowed such by those who candidly say what they think; provided they really think and reflect, and do not take things upon trust.

6. Yet in the calculus differentialis, which Method serves to all the same Intents and Ends with that of Fluxions.

Our modern Analysts consider:

• the Differences of finite Quantities
• the Differences of those Differences
• the Differences of the Differences of the first Differences.

And so on ad infinitum.

They consider Quantities infinitely less than the least discernible Quantity and others infinitely less than those infinitely small ones, and still others infinitely less than the preceding Infinitesimals, and so on without end or limit.

This leads to an infinite succession of Infinitesimals, each:

• infinitely less than the foregoing, and
• infinitely greater than the following.

As there are first, second, third, fourth, fifth, &c. Fluxions, so there are Differences, first, second, third, fourth, &c. in an infinite Progression towards nothing, which you still approach and never arrive at.

The most strange is that you can take a Million of Millions of these Infinitesimals.. Each of them is supposed infinitely greater than some other real Magnitude. You then add them to the least given Quantity, yet it shall be never the bigger.

For this is one of the modest postulata of our modern Mathematicians, and is a Corner-stone or Ground-work of their Speculations.