# Do we use 'with' or 'to' with 'proportionally'?

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?

• You can resolve the question by saying instead: "The temperature increases in proportion to pressure..." The adverb is inelegant here anyway. Commented Jul 3, 2017 at 7:30

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.

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):