This is a passage that has puzzled many readers!
“Vault” means “jump or leap, especially over or onto something”. The word is still used for various kinds of gymnastic feat, whether over equipment, or on the backs of horses.
So in these lines, Shakespeare gives us two metaphors in quick succession. In the first, Macbeth likens his intent (his plan to murder Duncan) to a horse and his ambition to the spur which is the only way to make the horse go forward. In the second, Macbeth likens his ambition to a rider jumping into the saddle of a horse, who leaps too high (tries to seize the crown of Scotland) misses his seat, and falls (meets with disaster).
I apprehend that there is not here one long-drawn metaphor, but two distinct ones. I have no spur to prick the sides of my intent; I have nothing to stimulate me to the execution of my purpose but ambition, which is apt to overreach itself; this he expresses by the second image, of a person meaning to vault into his saddle, who, by taking too great a leap, will fall on the other side.
Edmond Malone (1821). The Plays and Poems of William Shakspeare, volume XI, p. 81. London: F. C. and J. Rivington.
But difficulties arise if you attempt to reconcile the two metaphors into a single figure, for then you have to figure out how ambition can be both a spur and a rider. And if you have difficulty finding a satisfying combination, you might be tempted to suppose a misprint, insert a missing word, or invent a new sense of a word.
Here’s a selection of attempts by critics over the years to resolve these difficulties.
And here it may be permitted the Editor to profit by … correcting in Shakespeare what is absolute nonsense as now printed: “Vaulting ambition that o’erleaps itself”. It should be its sell. Sell is saddle, in Spenser and elsewhere, from the Latin and Italian.
Walter Savage Landor (1834). Citation and Examination of William Shakspeare, p. 65. London: Saunders and Otley.
It has been proposed (by whom we do not recollect) to read, instead of itself, its sell, its saddle. However clever may be the notion, we can scarcely admit the necessity for a change of the original. A person (and vaulting ambition is personified) might be said to overleap himself, as well as overbalance himself, or overcharge himself, or overlabor himself, or overmeasure himself, or overreach himself. There is a parallel use of the word over in Beaumont & Fletcher. “Prove it again, sir; it may be your sense was set too high, and so overwrought itself.” The word over in all these cases is used in the sense of too much. … After other Hanmer introduced side. The commentators say that the addition is unnecessary, inasmuch as the plural noun, sides, occurs just before. But surely this notion is to produce a jumble of the metaphor. Macbeth compares his intent to a courser: I have no spur to urge him on. Unprepared, I am about to vault into my seat, but I overleap myself and fall. It appears to us that the sentence is broken by the entrance of the messenger; that it is not complete in itself; and would not have been completed with side.
Charles Knight (1846). The Standard Edition of the Pictorial Shakspere, volume II, p. 16. London: Charles Knight.
I do not perceive any difficulty here, when we consider that the image in the Poet’s mind was that of horseman gallantly mounting into his seat. The words “spur” and “vault” plainly shew what was in the Poet’s mind. Macbeth says that it is no instigation from without, only the working of ambition within; the purposes of which are often defeated, as a person mounting a horse may take too high a leap, and so, instead of seating himself in the saddle, fall on the other side of the horse. The word “oft” seems lost before “o’erleaps,” and the word “side” is wanting to make the sense complete.
Joseph Hunter (1845). New Illustrations of the Life, Studies, and Writings of Shakespeare, volume II, p. 178. London: J. B. Nichols.
Here ‘other’ may be used substantively, as equivalent to ‘other side,’ which reading was given by Hanmer, and has support in the fact that it completes the quota of syllables for a perfect line. Perhaps ‘side’ was meant to be understood, with reference to the occurrence of the word in the preceding clause of the sentence.
Richard Grant White (1862). The Works of William Shakespeare, volume X, p. 520. Boston: Little Brown.
Malone’s explanation … does not assist us much; still less does the fanciful suggestion to read for “itself” its sell, i.e. its saddle. The only resolution of the enigma which presents itself to our mind is to suppose Intent and Ambition are represented in Macbeth’s disordered imagination by two steeds, the one lacking all incentive to motion, the other so impulsive that it overreaches itself and falls on its companion.
Howard Staunton (1864). The Works of William Shakespeare, volume IV, p. 315. London: Routledge.
As the text stands … we have in shadowy imagery a most extraordinary horse and rider. Macbeth was no more likely to wear a single spur that would strike on both sides than the Irishman was to discover the gun that would shoot round the corner. Moreover, his horse must have had three sides to it at the least. Now a horse may have four sides, right and left, inside and outside, and the street gamins will at times advise an awkward horseman to ride inside for safety, but it cannot have three sides. And if the single spur had pricked two sides, there could have been no other left for ‘vaulting ambition’ to fall on. The truth is that ‘sides’ is a misprint. The single spur of course implies a single side—the side of Macbeth’s intent, which leaves ‘the other’ for the ‘vaulting ambition’ to alight on in case of a somersault—the side of Macbeth’s unintent. The passage comes perfectly right if we read—‘To prick the side of my intent …’
Gerald Massey (1866). Shakspeare's Sonnets Never Before Interpreted: His Private Friends Identified, p. 599. London: Longmans.
Macbeth says he has nothing to goad him on to the deed,—nothing to stimulate his flagging purpose, like the private wrongs which he urges upon the murderers of Banquo,—but mere ambition, which is like one who, instead of leaping into the saddle, leaps too far and falls on the other side. The passage supplies a good example of confusion of metaphors. If the sentence be complete, ‘the other’ must be taken to mean ‘the other side,’ a not unnatural ellipsis, but one for which we can adduce no example. Hanmer reads ‘on the other side,’ which makes both sense and metre complete. Rowe prints ‘on th’other—,’ as if the sentence were interrupted by the entrance of Lady Macbeth. Mason conjectured ‘on the rider,’ and Bailey ‘on the earth’. For ‘itself’ in the previous line Singleton guessed ‘its sell’, i.e. ‘its saddle’.
William Clark & William Wright (1869). Macbeth, p. 99. Oxford: Clarendon.
Now by what aspect of the figure of equitation† shall this vacillation be set before us? Is it by the image of one getting on his horse, as the commentators explain; or of one mounted, and urging his steed towards the goal? The former might symbolise a single act of mind, but the case demands a figure that shall set forth a double, nay, a complex and conflicting state of motives. Moreover a person mounting clearly would not need any spurs either as instruments of or incitements to mounting. … No; Shakespeare’s hero is already mounted, and eager for the goal; but he and his steed are not of one mind, and for want of a spur, cannot be brought into harmony of action. To set forth the conflict between Macbeth’s ambition and his irresolution—whether this irresolution sprang from weakness, cowardice or conscience—what so apt as the two-fold image of an eager horseman on a balking horse, the spurless rider leaning forward (vaulting) on his laggard steed? This was the poet’s metaphor. … Manifestly we have here a figure of equitation. ‘Intent’ is the steed, ‘Ambition’ is the rider. The rider has no spur to prick on his halting steed, while he himself is urged forward by strong desires. Rising in his stirrups, and bending over (‘vaulting’), he overleaps himself. The goal of Macbeth’s ‘intent’ is the assassination; the goal of his ‘ambition’ is the throne. Macbeth must be a murderer before he can be a monarch. If the intent to murder halts, the desire to mount the throne will be futile. All this Macbeth knows and feels. He does not repine at any lack of ambition; that is in full force and action; it is o’erwrought: but over his purpose he mourns—that is infirm; over his courage—that needs ‘screwing to the sticking place’; and so, like an eager rider on a sluggish steed, he o’erleaps himself and ‘falls on the withers,’ and so Shakespeare wrote. … This reading seems to me to meet all the demands of the passage. Withers calls for no explanation, it explains itself. Whether copied by eye or ear, it was easy to mistake in sound or appearance ‘other’ for withers. This reading charges Shakespeare with no mixed or imperfect metaphors. It leaves his rhetoric and imagination unsuspect, brings the whole passage into harmony with itself and with the character of Macbeth—too ambitious to be innocent in thought, too cowardly to be guilty in deed. His imagination sicklied o’er with the pale cast of conscience, he is vacillating in purpose, irresolute in action, and querulous in speech.
Joseph Baugher Bittinger (1876). ‘What Shakespeare knew of horsemanship—a new reading of Macbeth I.7’. In Proceedings of the American Philological Association 1876, p. 40.
In my opinion, Malone is right that there are two metaphors here—no doubt the second metaphor was suggested or inspired by the first, but it is still a separate figure—and if you try to combine them and get something as absurd as Staunton’s show-jumping, or Bittinger’s theory that “vault” meant “lean forward” rather than “leap” and “other” was a misprint for “withers”, that is a sign that your criticism has o’erleaped itself.