I have been trying to read through Newton's Principia and this phase has come up several times so far, usually in the context of a mathematical operation. I cannot see any additional information or purpose in this phrase bring added, but it happens repeatedly. Did this phrase have a real purpose in the writing or is it just an artifact of the attempted direct translation from Latin?

Law III:[...] If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone -- Page 83

These things being done, we are to take the product (if I may so say) of the body A, by the chord of the arc, TA (which represents its velocity), that we may have its motion in the place A immediately before reflexion -- Page 91

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    All the answers so far seem to be interpreting or speculating on their own. But the phrase also seems to be a set phrase with a particular meaning special for the mid 1600's. Any answerer should try to address whether it was a set phrase at the time. – Mitch Jun 3 at 13:17
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    @mitch I wonder if it might mean "am I allowed to say this?" referencing whether the religious authorities of the time would permit this to be said. Life was different 400 years ago, and upsetting people in power was a bad idea. – Criggie Jun 4 at 5:59
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    @Criggie Life wasn't that different 400 years ago. They weren't cavemen. Upsetting people in power is still a bad idea, as we see daily. – Asteroids With Wings Jun 4 at 21:22
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    @AsteroidsWithWings there was certainly a difference in the authority of the church and what a scientist could say, comparing now and then. – Criggie Jun 4 at 22:52
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    @Criggie : nope, the church authority didn't care much about dictating what scientists are and aren't allowed to say, at least not as much as many people now assume. lesswrong.com/posts/rsAvZtTpTnHhmL5Ht/… – vsz Jun 5 at 5:37

This is a translation of a insertion that Newton made after the printing of the first edition. It is present in the second edition. You can see on the Cambridge facsimile, Newton has inserted the Latin phrase "ut ita dicam" in handwriting. This has been faithfully reproduced in the second edition.

This Latin phrase is discussed by an article which:

argues that its function is pragmatic in essence, since it operates both as a discourse marker, namely an interpretative device favoring the cooperative construction of the contextual meaning, and as a pragmatic marker capable of modulating the force of a potentially face-threatening statement or the employment of a daring term.

The function of this phrase is to engage the reader and ask them to cooperate with the writer and consider a potentially difficult concept (that the stone pulls on the horse)

"Ut ita dicam" may be translated as "So I say". "Ut ita" is a conjunction, roughly "so" and dicam is the first person singular subjective form of "dico: I say"

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I understand, if I may say so. That is the modern formatting of that phrase, to use an expression in an unusual way, and to declare that you're doing it. We could say "in a manner of speaking," or simply "if I can say it that way."

To pick it apart a bit,

the plane is held, if you will indulge me in what I do say to illustrate this ('if I may so' say) in the air by the negative air pressure created by the shape of its wings moving through the air.

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    Have you any supporting evidence that this is 'as it were', 'I'm struggling to find the right words here' rather than 'if I may be so bold as to assert'? (ie a language-orientated [easing the introduction of novel terminology, 'scare quotes'] pragmatic marker rather than an audience-orientated/persuasive pragmatic marker) – Edwin Ashworth Jun 3 at 11:44
  • @Araucaria-Nothereanymore. I'm not sure what you're referring to with 'fronting'. In "If I may be so bold...", 'so' is an adverb like 'very'. In "If I may say so...", 'so' is a pronoun replacing 'what I just said'. – Mitch Jun 4 at 12:35
  • @Araucaria-Nothereanymore. Oh. I didn't see that "('if I may so' say)". The punctuation confuses me so I can't really be sure that fronting is involved (and anyway it sounds really off to me to say "If I may so say") – Mitch Jun 4 at 12:43
  • The example about the airplane and wing shape, while a good grammatical example, is technically incorrect: it's a common but false myth. – vsz Jun 5 at 4:25

In both the examples, Newton appears to use the phrase to introduce an idea that was outside contemporary understanding. People had pressed stones with their fingers, and had their fingers pressed by stones, so there was no need to flag these ideas. People had seen horses dragging stones, but the idea that the stone was also dragging the horse (particularly if the horse was still moving forwards) was new. Multiplying a body by a chord was also a new and unintuitive idea.

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    I don't think Newton was contrasting the finger-stone and horse-stone examples, but rather basically saying they exhibit the same principles. – UuDdLrLrSs Jun 3 at 16:38

Long comment

I'm not an English speaker, so I give you my personal interpretation, based on my knowledge of history of Early Modern science.

IMO Newton uses it to highlight seemingly "paradoxical" statements, i.e. counter-intuitive statements, at least with respect to common sense and scientific knowledge of Newton's time.

Law III (page 14, Motte transl., Cajori edition, Chicago UP reprint).

We are accustomed to think that the horse pull the chariot and not that the chariot pull the horse in the reverse direction.

Coroll.III (page 18): "having not only lost its whole motion, but (if I may say so) one part more".

It sounds a little bit strange to loose more motion that "the whole".

Scholium to Coroll.VI (page 23): "These things being done, we are to take the product (if I may so say) of the body A, by the chord of the arc, TA (which represents its velocity)"

In classical mathematics, following the heritage of Ancient Greek mathematicians as exemplified by Euclid's Elements, it is allowed to multiply only homogeneous magnitudes: length with length, surface with surface, etc.

The modern algebraic approach (Renaissance, Descartes) was obviously known to Newton, but Newton decided not to use it into the Principia, using instead a "classical" geometrical language.

Thus, IMO, he is "begging pardon" for multiplying a body (mass) by a chord arc (a length representing velocity).

Prop.XVII (page 98): BD is not strictly speaking a diagonal because it does not connct two vertex of the equadrilateral figure ABFE.

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  • This is a clever analysis. But it's confirmed my thinking that we're using involved context to support the necessary reading of the expression, so involved that it requires detailed History of Science knowledge. It's off-topic here in some way. – Edwin Ashworth Jun 4 at 11:37
  • How exactly does this answer differ from the one given by @Hunter? – jsw29 Jun 4 at 22:15

I cannot comment on the rest of the Principia, but in the examples you have given here it appears to be akin to "by extension".

Ie. if x is true of fingers and stones, then, by extension, x is also true of a more complex situation where a horse is pulling a stone via rope.

"If I may so say" could be interpreted as "if I may so extrapolate/conject/assert".

He is essentially making the argument that his current statement follows from what was said before.

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