Understanding Ultimate Ratio of Equality

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SUMMARY

The discussion centers on the concept of the "ultimate ratio of equality," particularly in relation to the division of quantities into parts and their ratios as they approach infinitesimal values. The extract emphasizes that if two quantities are divided into equal parts, their ultimate ratios will remain consistent even as the number of parts increases and their magnitudes decrease. Additionally, the conversation touches on Newton's second law, F=ma, questioning whether a ball thrown in space will continue to accelerate or only do so momentarily, highlighting the need to understand the conditions required for sustained acceleration.

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  • Understanding of mathematical ratios and limits
  • Familiarity with Newton's laws of motion, specifically F=ma
  • Basic knowledge of calculus, particularly concepts of infinitesimals
  • Ability to interpret mathematical extracts and lemmas
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Students of mathematics and physics, educators explaining advanced concepts, and anyone interested in the foundational principles of ratios and motion in classical mechanics.

adiputra
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hi,
1)can anyone help me to understand ultimate ratio of equality? (from this extract)

Cor. Hence if two quantities of any kind are any how divided into an equal number of parts, and those parts, when their number is augmented, and their magnitude diminished in infinitum, have a given ratio one to the other, the first to the first, the second to the second, and so on in order, the whole, quantities will be one to the other in that same given ratio. For if, in the figures of this Lemma, the parallelograms are taken one to the other in the ratio of the parts, the sum of the parts will always be as the sum of the parallelograms; and therefore supposing the number of the parallelograms and parts to be augmented, and their magnitudes diminished in infinitum, those sums will be in the ultimate ratio of the parallelogram in the one figure to the correspondent parallelogram in the other; that is (by the supposition), in the ultimate ratio of any part of the one quantity to the correspondent part of the other.

2)if you are throwing a ball in a space and under equation F=ma, will the ball keep accelerating or it will accelerate for a momentary?



(sorry my english is very poor)
 
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For (1) it would help to have the lemma and the pictures if you could.

For (2) you should take a closer look at the equation F=ma. What is necessary to cause an acceleration? How do you apply such a thing?
 

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