Why does "footing the bill" mean "to pay"?

I hear people using the term footing the bill used to describe paying for something.

Why is the verb foot used to describe the meaning of paying?

From Etymology Online, footing (n.) as solid base for something evolved from the late 13c.:

"a base, foundation;" late 14c., "position of the feet on the ground, stance," a gerundive formation from foot (n.). Figurative meaning "firm or secure position" is from 1580s; that of "condition on which anything is established" is from 1650s.

From tinyonline.co.uk:

Footing was the act of adding up figures in a list and placing a total at the foot of the column. It was polite to ask a customer to foot the bill (check the arithmetic) as a euphemism for 'pay the bill'. In time, the euphemistic sense dropped away.

According to Today I Found Out:

The first written record of the phrase appears in 1819 in Estwick Evans Pedestrious Tour of Four Thousand Miles where the intrepid Evans noted, “My dogs . . . helped themselves to the first repast presented, leaving their master to foot the bills.”

In Report of the Superintendent of Common Schools of the State of New York, written in 1844, one sees it was clearly in use with that meaning:

Christine Ammer, The American Heritage Dictionary of Idioms (1994), has this entry for the expression:

foot the bill Pay the bill, settle the accounts [example omitted]. This expression uses foot in the sense of "add up and put the total at the foot, or bottom, of an account." {Colloq; early 1800s}

A Google Books search for the phrase finds an earliest publication date of 1841—followed by three examples from 1842. From "Superintendent's Report," in "Third Annual Report of the Directors and Superintendent of the Ohio Lunatic Asylum to the Fortieth [Ohio] General Assembly" (December 13, 1841):

They have been chiefly engaged, during the past season, in grading and arranging the ground in front of the asylum, which, by accurate measurement and calculation, they have found to contain exactly eight acres. And here is the amount of this labor, as estimated by a committee of themselves, which we should think exceedingly moderate, if we had to foot the bill : "As near as can be calculated, three acres of ground on the east half of lot in front of L. A., have been filled up to the average depth of nine inches, amounting to 3,637 cubic yards. ..."

From J.B. Turner, Mormonism in All Ages, or the Rise, Progress, and Causes of Mormonism (1842):

In 1835, the leaders found themselves some fifteen or twenty thousand dollars in debt for their temple, besides other expenses, and concluded to raise the wind again, by resorting to mercantile speculations. Accordingly, they ran the society into debt some one hundred thousand dollars, bought goods at the East, built steam-mills, bought farms, erected fine houses, &c., until the day of account drew near, when, of course, they found themselves bankrupt, and left their Mormon endorsers to foot the bill.

From William Goodell, The Rights and the Wrongs of Rhode Island (1842):

Thus bolstered, the monied and landed aristocracy, unlike the less wealthy majority, found means to raise the "sinews of war"—money—at least they could afford the risk of advancing it, from their own coffers, at a snug interest, in the prospect of making the State pay the expenses of their treasonable expedition against it — of taxing the disfranchized majority to foot the bill for the implements of their own butchery! No "plunder" nor incendiarism could be charged on the "respectable" allies of President Tyler, for that! For who but "fanatics" would suspect our slaveholding President of patronizing piracy?

And from an 1842 verse translation of La Fontaine's Fables:

There takes place nothing on this planet,/But fortune ends, whoe'er began it./In all adventures good or ill,/We look to her to foot the bill./Has one a stupid, empty pate,/That serves him never till too late?/He clears himself by blaming Fate.

In each of these instances (all of which are from U.S. authors), the group or entity footing the bill is not the one that incurred the expense; rather, there is a consistent sense that the "footing" is done on behalf of someone else. I think that today this remains an essential (or at least very common) element of the notion of "footing the bill"—and given that it appears in these very early instances of the phrase in print, I wonder how that notion arose from the neutral sense that Ammer gives of simply paying up or settling accounts for the bottom-line amount.

The earliest definition in print of the phrase is actually of the similar wording foot a bill, in Charles Mackay, Life and Liberty in America: or, A Tour in the United States and Canada in 1857–8 (1859):

To foot a bill, to sign or accept bill.

And Maximilian Scele De Vere, Americanisms; the English of the New World

To foot a bill, by paying the amount at the bottom of the account, is a phrase equally well known abroad and with us ...

But neither of these European authors seems to have noticed the sense of paying someone else's bill that attaches, even in the 1840s, to foot the bill.