"As we have no immediate experience of what other men feel, we can form no idea of the manner in which they are affected, but by conceiving what we ourselves should feel in the like situation. Though our brother is upon the rack, as long as we ourselves are at our ease, our senses will never inform us of what he suffers. They never did, and never can, carry us beyond our own person, and it is by the imagination only that we can form any conception of what are his sensations."
That I agree with.
"Whatever is the passion which arises from any object in the person principally concerned, an analogous emotion springs up, at the thought of his situation, in the breast of every attentive spectator."
That bit I'm not so sure about, but I find it interesting that Smith possibly identifies attention as a prerequisite for empathy. Those two quotes are from the book The Theory of Moral Sentiments, Part I, Section I, Chapter I by Adam Smith, published in 1759, quite a few years before the current fad of glorifying empathy as the psychological antidote to all of mankind's moral flaws and evil deeds.
The interesting thing about Smith is that he wrote at length about empathy and morality and he has also been identified by two professors of economics (Profs. Vernon Smith and Tyler Cowen) as possibly having been on the autistic spectrum. Debate that if you wish, but it is beyond dispute that Smith was in some ways quite a strange duck, which is true of many extremely smart people.
About Adam Smith
Cowen, Tyler (2010) The Age of the Infovore: Succeeding in the Information Economy. Plume, 2010.
[Apparently this is the same book as Create your own economy published in 2009 retitled. Vernon Smith, Sir Arthur Conan Doyle, Peter Mark Roget, Adam Smith, Hermann Hesse, Warren Buffett, Tim Page, Hikari Oe, Craig Newmark, Bram Cohen, Temple Grandin, Glenn Gould, Immanuel Kant, Thomas Jefferson are all discussed in this book with reference to the autistic spectrum]
Cowen, Tyler (2009) Create your own economy: the path to prosperity in a disordered world. Dutton, 2009.
[Adam Smith and many other famous people discussed in this book with reference to the autistic spectrum. The author of this book is an economics professor.]
Rae, John (1895) Life of Adam Smith. Macmillan & Co., 1985.
[see also Google Books for previews and various editions, no mention of AS or autism. Echopraxia episodes recounted on pages 157-158.]
Smith, Adam & Stewart, Dugald (editor) (1853) The theory of moral sentiments: or, An essay towards an analysis of the principles .... H. G. Bohn, 1853.
[The biographical essay about Smith by Stewart in this volume can be read through Google Books]
Smith, Vernon L. (2009) Rationality in economics: constructivist and
ecological forms.Cambridge University Press, 2009.
[In a footnote on page 19 Vernon Smith speculates that Adam Smith “may have had some of the earmarks of high-functioning autism or Asperger’s Syndrome”.]
Smith, Vernon L. (2008) Discovery - a memoir. Author House, 2008.
[brief speculation about Adam Smith and autism on page 190]
Stewart, Dugald (1973) ACCOUNT of the LIFE AND WRITINGS of ADAM SMITH, LL.D. From the Transactions of the Royal Society of Edinburgh [Read by Mr Stewart, January 21, and March 18, 1793] - Adam Smith, Glasgow Edition of the Works and Correspondence Vol. 3 Essays on Philosophical Subjects 
Edition used: Essays on Philosophical Subjects, ed. W. P. D. Wightman and J. C. Bryce, vol. III of the Glasgow Edition of the Works and Correspondence of Adam Smith (Indianapolis: Liberty Fund, 1982). Author: Adam Smith, Editor: J.C. Bryce, Editor: William P.D. Wightman
Part of: The Glasgow Edition of the Works and Correspondence of Adam Smith, 7 vols.
["The more delicate and characteristical features of his mind, it is perhaps impossible to trace. That there were many peculiarities, both in his manners, and in his intellectual habits, was manifest to the most superficial observer; but although, to those who knew him, these peculiarities detracted nothing from the respect which his abilities commanded; and although, to his intimate friends, they added an inexpressible charm to his conversation, while they displayed, in the most interesting light, the artless simplicity of his heart; yet it would require a very skilful pencil to present them to the public eye."]