p. 30: This Coalition of Numbers and Magnitudes being admitted, a plentiful Accession accrues to each Discipline. For it will be a very easy thing to discover and demonstrate very many theorems concerning Numbers by the Assistance of Geometry, which, by keeping within the common Limits of Arithmetic would scarcely, if at all, be capable either (p. 31) of Investigation or Demonstration: Also very many Things may be more briefly and clearly found out and demonstrated from hence. And reciprocally, the Ratio's or Reasons of Numbers being well understood will communicate not a little to the more evident Explication and strong Confirmation of many Geometrical Theorems. We will illustrate the Matter with an Example or two. It is an Arithmetical Theorem, that the Sum of an infinite (or indefinite) Series of Numbers increasing from Nothing to a certain term, which is the greatest according to the Ratio of the square Roots of Numbers continually exceeding one another by Unity, (i.e. as 0, 1, 2, 3, &. ad infinitum) is subsesquialter (2/3) the Sum of as many equal to the said greatest Term; Which Theorem I am of Opinion can never be exactly demonstrated by any Method in Arithmetic itself: but it is plainly deduced from Geometry. For if the Diameter of any Parabola be conceived to be divided indefinitely into many equal Parts, then the Right Lines which are ordinately applied to the Diameter, through the Points of the Divisions, will proceed in the same ratio, as is shewn in Geometry: But the Parabola which is constituted of these, whether Right Lines or Parallelograms, is there also demonstrated to be Subsesquialter (2/3) to the parallelogram, upon the same Base and of the same Height, or which is the same thing, to the Sum composed of as many Right Lines or Parallelograms equal to the greatest: From whence, the Agreement of Arithmetic with geometry being supposed which we desire to advance, it plainly follows, that a Series of Numbers of this sort is Subsesquialter (2/3) the Sum of as many equal to the greatest.

 Let there be a series, a0, ..., an, ..., where

  1. a0 = 0
  2. The series a0, ..., an, ... converges on some number Q (possibly infinite)
  3. the total number of terms is P (where P may be infinite)
  4. the ratio of an : amn : m.

In other words the series is, for some magnitude b (possibly infinitesimally small):

b*0, b*1, b*2, ..., b*n, ...

Let the limit of the series be Q.
Then, a0 + ... + an + ... = 2/3 P*Q
Or simply, as n => infinity,

b*0 + b*1 + b*2 + ... + b*n => 2/3 (n+1) *b*n

Let there be a parabolic segment ABC and a parallelogram (for convenience, a rectangle) AEFC with its base the base of the parabola and its top the tangent to the vertex of the parabola.  Let AC be the base of the parabolic segment and BD the altitude of the segment.  Hence,

Parallelogram AEFC = AC * BD

Parabola ABC = 2/3 parallelogram AEFC = 2/3 AC*BD 

Now we can conceive of the parabola as composed of lines parallel to the base such that b2x = y2 or alternatively y = bx.  Hence, if we take units (whatever we want to take as units) from the vertex (for y), we can see that the lines parallel to the base at these unit distances form a series,

b*0, b*1, b*2, ..., b*n, ...

As these units become infinitesimally small (we can vary b to keep the limit the same), we can conceive of the parabola as composed of the parallel lines.  In other words,

Parabola ABC            =   b*0 + b*1 + b*2 + ... + b*n + ... 
                                         t0        t1        t2                    tn
Parallelogram AEFC  =   AC  + AC  +  AC     + ... + AC       + ...

where each term AC of the parallelogram is matched up with a term b*n of parabola.  Hence the number of terms in the two series are the same.  Let the number of terms in each series be P.  Let AC = Q.  Since the parabola is composed of the b*n's and the parallelogram of the AC's, it follows that:

Parabola ABC = b*0 + b*1 + b*2 + ... + b*n + ... 
                        = 2/3 Parallelogram AEFC 
                         = 2/3 P*AC 
                         = 2/3 P*Q 

Likewise, we can take the parallelogram as composed of parallelograms and similarly the parabola.  Here, we can conceive of the parabola as composed of parallelograms with bases parallel to the base of the parabola such that c2x = y2 or alternatively y = cx.  Let d be the height of the parallelograms composing the parabola.  Hence,

Each paralleogram composing the parabola = d*cx

From this it follows that:

Parabola ABC           = d*c*0 + d*c*1 + d*c*2 + ... + d*c*n + ... 
                                      t0            t1             t2                    tn
Parallelogram AEFC =  d*AC  + d*AC  +  d*AC  + ... + d*AC       + ...

where each term d*AC of the parallelogram is matched up with a term d*c*n of parabola.  Hence the number of terms in the two series are the same.

  • Let the number of terms in each series be P. 
  • Let b = d*c, 
  • and let d*AC = Q 

Hence, since the parallelogram is composed of P d*AC's and the parabola of the parallelograms in it,

Parabola ABC = d*c*0 + d*c*1 + d*c*2 + ... + d*c*n + ... 
                        = b*0 + b*1 + b*2 + ... + b*n + ... 
                        = 2/3 Parallelogram AEFC 
                        = 2/3 P d*AC 
                        = 2/3 P*Q

                         = 2/3 P*AC 
                         = 2/3 P*Q 

Observe, that these analyses, both common in the 17th cent., play fast and loose with infinite sums.  Barrow appears not at all bothered by this (assuming that this analysis of his argument is correct).  In fact, Barrows theorem is false in modern Standard Analysis, since the number of lines composing the figures are not countable (cannot be enumerated in a series of terms, t0, t1, t2, ..., tn, ...), and one cannot speak of the height of the infinitely narrow parallelograms as other than 0.  Similarly, the series, b*0 + b*1 + b*2 + ... + b*n, ... either has as its limit 0 or 0, depending on the value of b.  In Robinson's, non-standard analysis the answer may be different.
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