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The temperature increases proportionally to/with pressure

I found some forums that used 'to' instead of 'with'. I have heard the adjective 'proportional' used with 'to', as 'X is proportional to Y'.

But when it comes to the adverb, I feel 'with' sounds better. But Google search (even though both expressions yield thousands of results, 'to' is more commonly used than 'with') and Ngrams disagrees with me. Any help?

  • 4
    You can resolve the question by saying instead: "The temperature increases in proportion to pressure..." The adverb is inelegant here anyway. – P. E. Dant Jul 3 '17 at 7:30
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OED has, for the phrase in proportion:

in (also †for, †of, †with) proportion: in a known or specific proportion to something else, or as something else varies; proportionately.

The inference is that the phrase in proportion takes the preposition to. This usage is supported by numerous examples, e.g.:

1990   Taxation & Environmental Policy (Inst. Fiscal Stud.) Comm. No. 19. 2 Taxing those who damage the environment in proportion to the damage they do.

The OED has, for the adjective proportional:

3. Math. That is in proportion; (of two or more pairs of quantities) having the same ratio.

Numerous supporting examples indicate that the adjective takes the preposition to, e.g:

1991   C. A. Ronan Nat. Hist. Universe 18/1 (caption) In Euclidean space, a circle's area is proportional to the square of its radius.

It would be expected that an adverb whose root adjective takes the preposition to would take the same preposition; yet the OED also has, for the adverb proportionally:

a. In a proportional manner or degree; in (due) proportion; correspondingly.

This usage is supported by numerous examples in which the adverb takes the preposition with, e.g.:

1960   A. H. Benade Horns, Strings, & Harmony vi. 110 The frequency of a string rises proportionally with the square root of the tension.

We are clearly on our own here, in the vast unfettered realm that is the living language. The OED would seem to have us use with, but there is no definitive statement to that effect. In the cited OED example provided for the adverb, it modifies the verb rise, which agrees in effect with the verb increase in your example. With a slight preference for with, I would choose the preposition that sounds best to me; either to or with will be understood by an English-speaking reader.

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Both prepositions "to" and "with" work here, although they have a different meaning.

The most common usage of "proportionally to" is within the scope of comparing something of one thing to something of another and means "to a proportionate degree" (Definition in Oxford Dictionary):

  • Your salary will rise proportionally to your workload. (Compare the rise of salary to the amount of workload)
  • Health problems increased proportionally to the number of hours worked. (Compare amount of health problems to the number of hours worked)

The most common usage of "proportionally with" is within the scope of either by means of something or due to the change of something:

  • Sevoflurane consumption does not decrease proportionally with fresh gas flows reduction. (A change in fresh gas flows reduction doesn't proportionally decrees Sevoflurane consumption)
  • Resizing a figure proportionally with handles. (Resizing a figure proportionally by means of handles)
  • Does expressive timing in music performance scale proportionally with tempo? (Does the change of tempo proportionally scale the expressive timing in music?)
  • 2
    "proptional(ly) to" is a set phrase. The examples of "proportionally with" are misleading, since the "with" is part of the next phrase, not joined to "proportionally". You could just as well (correctly) say "Resizing a figure proportionally using handles" or ".. decrease proportionally when fresh gas flows reduce" but those examples don't say anything general about how to use the phrases "proportionally using" or "proportionally when." - such "phrases" don't exist. – alephzero Jul 3 '17 at 15:53

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