That is not what Adam Smith wrote. He did not write anything about “as if by an invisible hand”.

From Adam Smith's Lost Legacy:

"Misreading Adam Smith, I hope Accidentally From the Princeton University press Blog, announcing a new economics title HERE:

“Adam Smith, Meet Captain Hook: The Upside of Pirate Greed” by Peter Leeson

“In 1776 Scottish moral philosopher Adam Smith published The Wealth of Nations. In it, he described the famed “invisible hand.” According to Smith, individuals pursuing their self-interests are led, “as if by an invisible hand,” to promote others’ interests as well.

Your grocer, for example, wants to serve his own interest—he wants to make money. But to do so he must serve your interest as well. He must provide you with the highest quality groceries at the lowest possible price or you’ll patronize a competitor that does instead. The grocer doesn’t care about you, of course; he doesn’t even know you. He cares about himself, but in serving himself he serves you too.”

Comment
Peter Leeson, BB&T Professor for the Study of Capitalism at the Mercatus Center at George Mason University and author of the new book, The Invisible Hook: The Hidden Economics of Pirates and he also blogs at The Austrian Economists, is almost right and spoils the accuracy of his interpretation of the grocer’s pursuit of his self-interest in serving her customer’s self-interest, which in turn serves hers.

Yes, we serve our self-interest best in exchange by serving the interests of others.
But, I am not sure of his accuracy of interpretation in his first paragraph:

“According to Smith, individuals pursuing their self-interests are led, “as if by an invisible hand,” to promote others’ interests as well.”

That is not what Adam Smith wrote. He did not write anything about “as if by an invisible hand”. The quotation marks around this statement suggests that Adam Smith wrote those words; he most certainly did not.

There is no ‘as if’ attached to ‘led by an invisible hand’. Peter Leeson has added them and incorrectly wrapped them in quotation marks.

Also the way the sentence is written implies that Adam Smith was making a general statement applying to all individuals in all cases; he wasn’t.

He wrote about a specific set of individuals (some but not all wholesale merchants) whose risk aversion to foreign trade led them to prefer to deploy their capitals in their domestic locality, despite the higher monopoly-driven profits from trading with the British colonies in North America under the protection of the Navigation Acts, enforced by the Royal Navy.

These individual decisions meant domestic annual product was higher than it otherwise would be, and because the whole is the sum of its parts, the domestic economy, and resulting employment and output, were greater than they would otherwise be.

Having explained all this clearly and adequately, Smith summarises his explanation, with the well-known 18th-century literary metaphor of ‘led by an invisible hand’ (WN IV.ii.9: 456), which applied in this one specific case and ‘in this, as in many other cases’, but not all cases, as the over 70 examples he mentions along the way of self-interests not being beneficial to society in Books I, II and III of Wealth of Nations.

In fact, Smith analyses how prices are determined, how markets work, and how the ‘great orders’ go about their business without mentioning ‘an invisible hand’ at all.

Professor Peter Leeson should have written his sentence as:

“According to Smith, SOME individuals pursuing their self-interests are led IN MANY BUT NOT ALL CASES, “by an invisible hand,” to promote others’ interests as well.”

By generalising Smith’s thinking in the manner he did, Professor Leeson repeats the mantra of some ultra-conservative-minded propagandists for State Corporate Capitalism( I AGREE WITH THIS LABEL. ) (which they are perfectly entitled to assert in their own names), but their assertions are their own and have nothing to do with Adam Smith."

The same thing happens with Burke.