Where to start and end an aliyah

(6) On each day of Hanukkah they read selections from the portion of the dedication of the altar by the tribal princes (Numbers 7). On Purim they read the portion of “And Amalek came” (Exodus 17:8–16). On the New Moon they read the portion of “And in the beginnings of your months” (Numbers 28:11–15). And in the non-priestly watches they read the act of Creation (Genesis 1:1–2:3). The Jewish people were divided into twenty-four watches. Each week, it would be the turn of a different watch to send representatives to Jerusalem to be present in the Temple to witness the sacrificial service. Those remaining behind would fast during the week, from Monday to Thursday, offer special prayers, and read the account of Creation from the Torah. On fast days, they read the portion of blessings and curses (Leviticus, chapter 26). One should not interrupt the reading of the curses by having two different people read them. Rather, one person reads all of them. On Mondays, and on Thursdays, and on Shabbat during the afternoon service, they read in accordance with the regular weekly order, i.e., they proceed to read the first section of the Torah portion that follows the portion that was read on the previous Shabbat morning. However, these readings are not counted as a progression in the reckoning of reading the Torah portions, i.e., they do not proceed on Monday to read the section that immediately follows the section read on Shabbat during the afternoon, and then the following section on Thursday. Rather, until the reading on the following Shabbat morning, they return to and read the same first section of the Torah portion that follows the portion that was read on the previous Shabbat morning. On Festivals and holidays, they read a portion relating to the character of the day, as it is stated: “And Moses declared to the children of Israel the appointed seasons of the Lord” (Leviticus 23:44), which indicates that part of the mitzva of the Festivals is that the people should read the portion relating to them, each one in its appointed time.

(2) On the days of the New Moon and on the intermediate days of a Festival, four people read from the Torah; one may neither decrease the number of readers nor add to them. And one does not conclude with a reading from the Prophets. Both the one who begins the reading and the one who concludes the reading from the Torah recite a blessing. The first reader recites a blessing before the beginning of the reading, and the last reader recites a blessing after its conclusion, but the middle readers do not recite a blessing. The mishna formulates a general principle with regard to the number of people who read from the Torah on different occasions. This is the principle: Any day on which there is an additional offering sacrificed in the Temple and that is not a Festival, i.e., the New Moon and the intermediate days of a Festival, four people read from the Torah; on a Festival, five people read; on Yom Kippur, six people read; and on Shabbat, seven people read. One may not decrease the number of readers, but one may add to them. And on these days one concludes with a reading from the Prophets. Both the one who begins the reading and the one who concludes the reading from the Torah recite a blessing; one recites before the beginning of the reading and one recites after its conclusion, but the middle readers do not recite a blessing.

(3) The Torah is never read in public in the presence of fewer than ten adult free men.1See Chapter 8, Halachot 4-6. No fewer than ten verses are read.2See Halachah 1. Hagahot Maimoniot notes that the reading for Purim contains only nine verses, but explains that since this passage discusses a complete subject, an exception is made. (See also Shulchan Aruch, Orach Chayim 137:1.) Vayedaber3i.e., a verse beginning "And God said to Moses:...." Although it is merely an introductory phrase, it... is counted as one of them.4i.e., one of the required ten verses. The Shulchan Aruch (Orach Chayim 137:4) states that, after the fact, if only nine verses were read, it is sufficient. No fewer than three men should read.5See Halachah 1.
[When] beginning a passage6In addition to the 54 weekly Torah portions, the Torah is divided into 290 smaller passages (parshiot). (See Hilchot Sefer Torah, Chapter 8.) from the Torah, [one should read] at least three verses,7A person who begins a new passage in the midst of his aliyah should read at least three verses, lest another person enter when he begins that passage and think that he has read fewer than three verses (Megillah 22a).

The 14Shulchan Aruch (Orach Chayim, loc. cit.) states that if a person read only two verses, they must be read again. and one should not conclude less than three verses from the conclusion of a passage.8lest someone who leaves before the next person reads from the Torah come to the mistaken conclusion that he has read fewer than three verses (Megillah, loc. cit.). (See Chapter 13, Halachah 4. See also Shulchan Aruch, Orach Chayim, 138.) Each reader should not read fewer than three verses.9The three verses allude to the threefold division of the written law (Megillah 24a). Even when an entire passage has only two verses, the reader must add at least three more verses from another passage.
From this halachah, it appears that the custom of reading the first aliyah of the portion to be read on the following Sabbath on Mondays and Thursdays was not fixed in the Rambam's time. Hence, these ground rules were necessary. (See Mishnah Berurah 137:4.)

(1) To Not Leave Over Less Than 3 Verses In a Parsha. Containing 1 Se'if:
The one who reads from the Torah may not leave over any less than 3 verses in a paragraph because [we are concerned for] those who leave the synagogue then, that they will say that the oleh (i.e. reader) after him will only read the two remaining verses [of the paragraph]. And similarly, one may not begin less than 3 verses into it (i.e. a paragraph) because of those who enter the synagogue then, that they will say that the first [reader] only read 2 verses. Rem"a: And there is no difference between an open paragraph and a closed paragraph. And a paragraph that only has two verses: it is permitted to leave over [these two verses] at the beginning [of a paragraph] and to stop there (T'rumat HaDeshen, siman 22). And one should aim to always begin to read on a good matter and to end on a good matter (Or Zarua and Meimoni, Chapter 13 of Hilkhot T'fillah). And see below, Chapter 428.