Making up a tefillah

The Gemara on 26a–b discusses the laws of tashlumin, making up a prayer you had missed by saying shemoneh esrei twice at the next prayer.

The Gemara raises the issue as a challenge to the fixed times of prayer:

וכולי עלמא עד חצות ותו לא? והאמר רב מרי בריה דרב הונא בריה דרבי ירמיה בר אבא אמר רבי יוחנן: טעה ולא התפלל ערבית - מתפלל בשחרית שתים, שחרית - מתפלל במנחה שתים! - כולי יומא מצלי ואזיל, עד חצות - יהבי ליה שכר תפלה בזמנה, מכאן ואילך - שכר תפלה יהבי ליה, שכר תפלה בזמנה - לא יהבי ליה.

Soncino:

And may other people delay till midday, but no longer? Has not R. Mari the son of R. Huna the son of R. Jeremiah b. Abba said in the name of R. Johanan: If a man erred and did not say the evening Tefillah, he says it twice in the morning. [If he erred] in the morning, he says it twice in the afternoon? — He may go on praying the whole day. But up to midday he is given the reward of saying the Tefillah in its proper time; thereafter he is given the reward of saying Tefillah, but not of saying Tefillah in its proper time.

By the way, right before that passage, Soncino translates ותיקין as “wathikin.”

The Gemara goes on to cite opinions than any prayer can be made up at the next prayer, whether it's Minchah at Arvit, or even an Amidah of a weekday made up on Shabbat and vice versa. The Gemara also establishes that these makeups are only for mistakes, not intentional misses:

מיתיבי: +קהלת א'+ מעות לא יוכל לתקן וחסרון לא יוכל להמנות; מעות לא יוכל לתקן - זה שבטל קריאת שמע של ערבית וקריאת שמע של שחרית, או תפלה של ערבית, או תפלה של שחרית; וחסרון לא יוכל להמנות - זה שנמנו חביריו לדבר מצוה ולא נמנה עמהם! - אמר רבי יצחק אמר רבי יוחנן: הכא במאי עסקינן - שבטל במזיד. אמר רב אשי: דיקא נמי דקתני בטל ולא קתני טעה - שמע מינה.

An objection was raised: That which is crooked cannot be made straight, and that which is wanting cannot be numbered. ‘That which is crooked cannot be made straight’; this applies to one who omitted the Shema’ of the evening or the Shema’ of the morning or the Tefillah of the evening or the Tefillah of the morning. ‘And that which is wanting cannot be numbered’: this applies to one whose comrades formed a group to perform a religious act and he was not included with them. — R. Isaac said in the name of R. Johanan: With what case are we dealing here? With one who omitted deliberately. R. Ashi said: The proof of this is that it says ‘omitted’, and it does not say, ‘erred’. This proves it.

The Yalkut Bi'urim in Mesivta goes on for twenty daf-sized pages, in small type, of iyun on the issue of tashlumin. Here's some interesting points I picked out.

What's the reason that you can make up a tefillah? Many mefarshim (Tosafot, Talmidei Rabbeinu Yonah, Rosh, Levush) explain that it's because tefillah is for mercy, and is relevant all day long—"would that man would pray the entire day"—you can make up your tefillah at another time. This is why Musaf cannot be made up: its purpose isn't mercy, but to stand in for a sacrifice, which has the principle of עבר זמנו בטל קרבנו, if the time runs out the sacrifice is lost.

But wouldn't that explanation mean you could make up a tefillah even if you skipped it intentionally? The Gemara says that intentionally skipping gives you something "crooked" that "cannot be made straight." It sounds like you have no option of tashlumin. But if you can pray voluntarily at any time, why can't you make up the prayer voluntarily?

Well, indeed you can. So said Rav Hai Ga'on, as quoted by the Semak, Rashba, Rosh, and Tur. (I didn't know that the Semak was the earliest of that list. He was a late French Tosafist who lived from about 1210 to 1280. Was ten years younger than the Semag, also French.) As for the crookedness problem, the Penei Yehoshua explains that the concept of tashlumin corresponds to three levels of sacrifice. The tefillah in its proper time is like the tamid. If you miss that you are obligated to bring an olah. And if you miss the prayer intentionally the best you can do is a voluntary olah. It's still "crooked" because you bring it without an obligation.

There's a lot more detail to discuss. Do you have to say tashlumin in the timeslot of the next prayer or can you say it any time—for example, can you make up Shaharit right before minhah gedolah? Can you make it up two timeslots later, e.g. making up Shaharit at Arvit? What happens when the current timeslot adds something to the Amidah, like havdalah or ya'aleh ve-yavo? The iyun gets very interesting.

I'll end by noting an interesting tension here regarding the basis of the tefillot. On the one hand, the times that regiment our prayers come from the sacrifices, which have a strict schedule. But on the other hand, we have this flexibility of mercy, perhaps reflecting the basis of tefillah in the avot. The problem with this idea is that the basis in the avot is only according to one opinion in the mahloket on 26b. But at least within that opinion it sounds like a nice hakirah.