People gather for a vigil at Whole Women’s Health Clinic in McAllen, Texas, on March 6, 2014. Photograph by Jennifer Whitney/The New York Times via Redux

How do you count women in Texas, and when do the numbers get big? There is a good deal of bad math in a decision made last week, by the Fifth Circuit Court of Appeals, that had the effect of closing all but eight abortion clinics in the state; until recently, there were about forty. Five million four hundred thousand Texans are women of childbearing age. Almost one and a half million of them will live more than a hundred miles from any clinic; nine hundred thousand will live more than a hundred and fifty miles away, seven hundred and fifty thousand more than two hundred and fifty miles. For a good many, there will be more than five hundred miles to go, unless they want to cross the border and take their chances in Mexico. For a two-to-one majority on the Fifth Circuit panel, that just wasn’t enough women for them to worry about.

The Texas clinics will close because of a law, passed by the state legislature last year, that placed new regulations on clinics that provide abortions. The Supreme Court has found that women cannot be cheated of their right to end a pregnancy before viability by way of laws that place an “undue burden” on them, as standard laid out in Planned Parenthood v. Casey, in 1992. But, as Jeffrey Toobin recently wrote, courts in recent years have become increasingly merciless in what they consider undue for a woman at what is often a moment of profound crisis, to the point where almost no burden seems too heavy.

Several aspects of the new law, like one requiring doctors to have admitting privileges at hospitals within a certain distance, survived challenges. But, in August, the District Court Judge Lee Yeakel struck down a rule that clinics have to be outfitted and operated as ambulatory surgical centers, even if they only provided medication-induced abortions early in pregnancies. Yeakel’s decision came after a trial at the District Court level that included testimony that requirement was not practical for most clinics, would leave no clinics open south or west of San Antonio, and was not based on any sound medical rationale. The state wanted the provisions to go into effect regardless, pending its appeal; Yeakel said no. The appeals court has lifted that stay, saying that it thought the law would ultimately survive the challenge. (It did leave room for a partial reprieve for a clinic in El Paso, though not for one in McAllen.) And so, on Friday, thirteen clinics in Texas began turning patients away.

The Fifth Circuit judges picked up on another phrase in Casey: “a large fraction.” A way to tell if a burden is undue is if it presents obstacles for a large fraction of the women for whom it is relevant. The fraction the Fifth Circuit calculated was one-sixth: nine hundred thousand women who would have to travel more than a hundred and fifty miles out of five million four hundred thousand who could possibly get pregnant—“not a large enough fraction to impress the appeals court,” as Ruth Marcus put it, no matter the absolute number. There are, if one is counting, at least three reasons this logic is wrong.

First, a sixth can be pretty large, depending on what the numerator (one, in this case) and the denominator (the six) represent. (One-sixth of New York City’s population lives in the Bronx.) That is why one uses a word like large rather than something more definite, like majority. When it comes to a decision that can shape a woman’s life, this Texas sixth is a large fraction—and that alone should have been enough for the judges.

Second, it’s not clear at all that the majority chose the right numerator or denominator—that the fraction really is a sixth. First, the numerator: Is it only the women who have to drive these distances who are affected when a state that, until recently, had sixty-to-seventy-two thousand abortions each year, suddenly has only eight clinics—all in a few cities? Or does it also mean that the women in the next clinic over will soon find it hard or impossible to get an appointment? Speed matters a great deal for abortion; Texas’s law also included a twenty-week limit. (In another sign of fractional bad faith, the majority suggested that a woman who had been a hundred and fifty miles from a clinic and was now two hundred and fifty miles away might only be facing an “incremental increase of 100 miles.”)

One can also reconsider the denominator, the bottom number. In Casey, the Supreme Court upheld some restrictions in Pennsylvania but overturned a requirement that married women notify their husbands. The state of Pennsylvania had argued that only twenty per cent of women seeking abortions were married and that ninety-five per cent would tell their husbands anyway, and so the fraction affected was tiny—maybe one per cent, and therefore too few to count. The Court rejected that math, saying,

The analysis does not end with the one percent of women upon whom the statute operates; it begins there. . . . The proper focus of constitutional inquiry is the group for whom the law is a restriction, not the group for whom the law is irrelevant.

The denominator that the Court chose in that case was “married women seeking abortions who do not wish to notify their husbands of their intentions and who do not qualify for one of the statutory exceptions to the notice requirement.” The fraction affected was suddenly very large.

The Texas decision briefly looks at the argument for a different denominator—women whose options will get worse because of the law—but then rejects it, bizarrely enough, because the resulting fraction is too large: it “would make the large fraction test a tautology, always resulting in a large fraction.” But that is only true if the burden on women for whom the law is relevant is, indeed, undue. One can imagine a law that presented X women with obstacles that Y of them could, nonetheless, easily navigate. What the judges see as a “tautology” is a sign that something is seriously wrong with the Texas law.

Third, as the dissenting judge in the Texas case noted, Casey doesn’t just talk about fractions: it talks about a “significant number” of women who, under the spousal-notification requirement, would not have meaningful access to abortion. After reviewing statistics on domestic violence, the Casey decision notes,

We must not blind ourselves to the fact that the significant number of women who fear for their safety and the safety of their children are likely to be deterred from procuring an abortion as surely as if the Commonwealth had outlawed abortion in all cases.

When does nine hundred thousand seem like an insignificant number of women?

There is another factor, involving other numbers: poverty. The Fifth Circuit judges acknowledged that women without much money would be more affected by the law than others: they might not have a car, or a way to take a day off from work to drive six hours. But that didn’t, somehow, change the judges’ calculation.