A new antiabortion TV ad appeared last week, just in time for the inauguration of a president whose support for abortion rights is unqualified. The ad shows the ultrasound image of a fetus in the womb. As the camera slowly moves in, a message gradually appears onscreen:

This child’s future is a broken home.

He will be abandoned by his father.

His single mother will struggle to raise him.

Despite the hardships he will endure…

this child will become…

the first African-American president.

Then, alongside a picture of President Obama, comes the closing message: “Life: Imagine the Potential.”

Last week also brought the 36th anniversary of the Supreme Court decision legalizing abortion, and with it the annual March for Life on Jan. 22. Tens of thousands of Americans, most of them in their teens and 20s, gathered in Washington to implore the new president to help end “the intentional killing of an estimated 3,000 pre-born boys and girls each day,” in the words of an open letter on the March for Life website. For his part, Obama issued a statement restating his support for abortion rights and insisting that Roe v. Wade “protects women’s health and reproductive freedom.”

Endlessly, the abortion battle goes on. The absolutists are forever polarized, but most Americans want to have it both ways. In poll after poll, substantial majorities say that abortion should be legal in only limited circumstances, if not banned outright. Only about one voter in five wants abortions to be legal at any time for any reason - i.e., abortion on demand. Yet by equally clear majorities, Americans say that they support Roe and would not want it overturned.

But these are irreconcilable positions.

Contrary to popular belief, the Supreme Court did not allow states to ban late-term abortions or restrict abortion on demand only to the first three months of pregnancy. It is true that Roe appeared to do so. Justice Harry Blackmun’s majority opinion declared that states could not regulate abortion at all in the first trimester and could do so thereafter only “in ways that are reasonably related to maternal health.” Once a fetus became viable, Blackmun wrote, states could regulate and even prohibit abortion, “except where it is necessary, in appropriate medical judgment, for the preservation of the life or health of the mother.”

Those 20 words became the exception that swallowed the rule.

Roe wasn’t the only abortion case the court decided on Jan. 22, 1973. In a companion case, Doe v. Bolton, the justices decided that “medical judgment may be exercised in the light of all factors - physical, emotional, psychological, familial, and the woman’s age - relevant to the well-being of the patient. All these factors may relate to health. This allows the attending physician the room he needs to make his best medical judgment.”

Taken together, Roe and Doe meant that abortion could not be barred at any stage of a pregnancy. The “attending physician” could always say that in his medical judgment, the woman’s “emotional” or “familial” health made it necessary to abort her unborn child. The result has been 36 years of abortion on demand at any stage of pregnancy.

Americans can be forgiven for not realizing what Roe really wrought. It has never been easy for its supporters to acknowledge its true impact. Chief Justice Warren Burger, who concurred in the decision, was sure that abortions would be performed only “on the basis of carefully deliberated medical judgments,” not merely for reasons of convenience. “Plainly,” he wrote, “the Court today rejects any claim that the Constitution requires abortion on demand.”

Burger was wrong, but he wasn’t alone. Right from the start, the media have gotten it wrong, too. The morning after the decision, The New York Times reported on Page 1 that the high court had “overruled today all state laws that prohibit or restrict a woman’s right to obtain an abortion during her first three months of pregnancy.” That mistake has been repeated endlessly in the 36 years since.

Since 1973, more than 40 million US pregnancies have ended in abortion: Ours is the most liberal abortion culture in the advanced world. Reasonable people can differ over whether to preserve Roe or overrule Roe. But surely the way to begin is to understand Roe.